.ELEMENTS 


OF 


MACHINE   DESIGN- 


BY 


DEXTER  S.  KIMBALL,  A.B. 


Professor    of    Machine     Design    and     Construction,     Sibley     College,   Cornell    University, 

Formerly  Works    Manager,   Stanley  Electric  Manufacturing  Company. 

Member  of  the  American  Society  of  Mechanical  Engineers. 


AND 


JOHN  H.  BARR,  M.S.,  M.M.E. 

Manager,    Smith   Premier  Works.      Formerly  Professor  of  Machine  Design,  Sibley  College, 
Cornell  University.      Member  of  the  American  Society  of  Mechanical  Engineers. 


OF   THE 

UNIVERSITY 

OF 


FIRST  EDITION 
FIRST  THOUSAND 


NEW  YORK 

JOHN   WILEY  &   SONS 

LONDON:      CHAPMAN  &  HALL,   LIMITED 
1909 


COPYRIGHT    1909 

BY 
DEXTER  S    KIMBALL  AND  JOHN   H.    BARR 


Electrotyped  and  Printed  by  Publishers  Printing  Co..  New  York,  U.  S.  A. 


PREFACE 

THIS  book  is  the  outgrowth  of  the  experience  of  the  authors 
in  teaching  Machine  Design  to  engineering  students  in  Sibley 
College,  Cornell  University.  It  presupposes  a  knowledge  of 
Mechanism  and  Mechanics  of  Engineering.  While  the  former 
subject  is  a  logical  part  of  Machine  Design,  it  may  be,  and  usually 
is,  for  convenience,  treated  separately  and  in  advance  of  that 
portion  of  the  subject  which  treats  of  the  proportioning  of  machine 
parts  so  that  they  will  withstand  the  loads  applied.  The  same 
logical  order  is  usually  followed  in  actual  designing,  as  it  is, 
ordinarily,  necessary  and  convenient  to  outline  the  mechanism 
before  proportioning  the  various  members. 

With  the  mechanism  determined,  the  remainder  of  the  work 
of  designing  a  machine  consists  of  two  distinct  parts: 

(a)  Consideration  of  the  energy  changes  in  the  machine,  and 
the  maximum  forces  resulting  therefrom. 

(b)  Proportioning  the  various  parts  to  withstand  these  forces. 

This  logical  procedure,  and  the  fundamental  principles  under- 
lying the  first  part  (a),  are  seldom  made  clear  to  the  student, 
in  works  of  this  character;  and  such  information  as  is  given  on 
energy  transformation  in  machines  is,  in  general,  that  relating 
to  special  cases  or  types.  A  thorough  understanding  of  these 
general  principles  is,  however,  in  most  cases,  essential  to  success- 
ful design,  since  a  consideration  of  the  machine  as  a  whole 
necessarily  precedes  consideration  of  details.  A  very  brief 
discussion  of  typical  energy  and  force  problems  is  given,  there- 
fore, in  Chapter  II,  in  the  hope  of  making  this  important  matter 
somewhat  clearer  to  the  beginner. 

While  the  treatment  presented  presupposes  a  knowledge  of 
Mechanics  of  Materials,  a  brief  discussion  of  the  more  important 
straining  actions  is  given  in  Chapter  III,  partly  to  make  the  appli- 
cation of  the  various  formulae  to  engineering  problems  somewhat 

iii 


iv  PREFACE 

more  definite,  and  partly  to  present  such  rational  theory  as  is  of 
assistance  in  selecting  working  stresses  and  factors  of  safety. 
This  discussion  serves  also  to  show  why  certain  equations  have 
been  selected  in  preference  to  others,  and  also  to  collect  in  concise 
form  the  more  important  equations  relating  to  stress  and  strain 
with  which  the  designer  needs  to  be  familiar. 

The  general  principles  of  lubrication  and  efficiency  are 
discussed  in  Chapter  IV.  Both  of  these  are  of  prime  importance 
to  the  engineer;  and  while  the  discussion  is  necessarily  brief  it  is 
believed  that  the  fundamental  principles  are  fully  covered. 

The  remainder  of  the  book  is  devoted  to  the  discussion  of 
some  of  the  more  important  machine  details,  with  a  view  of 
showing  how  the  theoretical  considerations  and  equations  dis- 
cussed in  the  first  part  of  the  work  are  applied  and  modified  in 
practice.  The  treatise  is,  in  no  sense,  a  hand-book,  neither  is  it 
a  manual  for  the  drafting  room,  but  is  a  discussion  of  the  funda- 
mental principles  of  design,  and  only  such  practical  data  have  been 
collected  as  are  needed  to  verify  or  modify  logical  theory.  It  is 
hoped  that  the  illustrative  numerical  examples  which  are  intro- 
duced throughout  the  work  may,  in  conjunction  with  the  analy- 
tical methods  given,  suggest  proper  treatment  of  practical  prob- 
lems in  design.  The  treatment  of  all  topics  is  necessarily  brief, 
as  it  was  desired  to  obtain  a  text-book  which  could  be  conve- 
niently covered  in  one  college  year  and  yet  present  the  salient 
features  of  the  subject  needed  by  the  student  as  a  preparation 
and  basis  for  more  advanced  work.  While  intended  primarily  for 
engineering  students  it  is  hoped  that  it  may  also  prove  of  some 
interest  to  the  practising  designer.  It  has  been  the  endeavor  in 
the  preparation  of  the  book  not  only  to  develop  rational  analytical 
treatment,  with  due  regard  to  constructive  considerations  and 
other  practical  limitations,  but  to  reduce  the  analysis  to  such 
forms  and  terms  that  definite  numerical  results  can  be  obtained 
in  concrete  problems. 

Considerable  of  the  matter  contained  in  the  book  has  already 
been  published,  specially  for  the  use  of  students  in  Sibley  College, 
under  the  title  of  "  Special  Topics  on  the  Design  of  Machine 
Elements,"  by  John  H.  Barr,  and  also  in  "Elements  of  Machine 


PREFACE  v 

Design,"  Part  I,  by  the  Authors.  The  writers  have  availed  them- 
selves freely  of  the  work  of  many  others  in  the  field,  for  which 
due  credit  is  given  in  the  text. 

The  authors  are  especially  indebted  to  Professor  G.  F.  Blessing 
of  Swarthmore  College,  Professors  W.  N.  Barnard,  L.  A.  Darling, 
and  C.  D.  Albert  of  Sibley  College,  Cornell  University,  all  of 
whom  have  given  instruction  in  the  course  at  various  times,  and 
also  to  Mr.  A.  J.  Briggs,  for  many  helpful  suggestions  and 
criticisms.  They  will  be  very  grateful  for  further  suggestions 
or  criticisms  which  will  improve  the  book. 

D.  S.  K. 
J.  H.  B. 

ITHACA,  N.  Y.,  June,  1909. 


CONTENTS 

CHAPTER   I 

ON 

OF   MACHINE   DESIGN, 


PAGE 

INTRODUCTORY.      DEFINITIONS   AND   FUNDAMENTAL  PRINCIPLES 


CHAPTER   II 

THE  ENERGY  AND  FORCE  PROBLEM.    CONSIDERATION  OF  MACHINES 
AS  A  MEANS  OF  MODIFYING  ENERGY,    .        .        •.        .        .      6 

CHAPTER   III 

STRAINING   ACTIONS   IN   MACHINE   ELEMENTS.      FUNDAMENTAL 
FORMULAS  FOR  STRENGTH  AND  STIFFNESS,     .        .  "     .        .31 

CHAPTER   IV 

FRICTION,  LUBRICATION,  AND  EFFICIENCY,    .       ....        •        .96 

CHAPTER  V 
SPRINGS,        .        .        .        ...        .        .        .        .        .114 

CHAPTER  VI 
RIVETED  FASTENINGS,    .  .  136 

CHAPTER  VII 

SCREWS   AND   SCREW   FASTENINGS,       .  .  .  156 

CHAPTER  VIII 

KEYS,   COTTERS,  AND  FORCE  FITS,     .        .        .  .  t     .  190 

CHAPTER   IX 

TUBES,  PIPES,  FLUES,  AND  THIN  PLATES    .        .        .  .211 

vii 


viii  CONTENTS 

CHAPTER   X 

PAGE 

CONSTRAINING  SURFACES,  SLIDING  SURFACES,  JOURNALS,  BEARINGS, 
ROLLER  AND  BALL  BEARINGS, 232 

CHAPTER   XI 
AXLES,  SHAFTING,  AND  COUPLINGS, 285 

CHAPTER   XII 
BELT,  ROPE,  AND  CHAIN  TRANSMISSION, 309 

CHAPTER  XIII 

APPLICATIONS  OF  FRICTION.    FRICTION  WHEELS,  FRICTION  BRAKES, 
AND  CLUTCHES, •       .        .        .        .  350 

CHAPTER  XIV 

TOOTHED  GEARING,  SPUR,  BEVEL  AND  SCREW  GEARS,    .        .        .  364 

CHAPTER  XV 

FLYWHEELS,  PULLEYS  AND  ROTATING  Discs,     .     .        .        .      406 

CHAPTER  XVI 

MACHINE  FRAMES  AND  ATTACHMENTS,        .  428 


MACHINE    DESIGN 


CHAPTER  I 
INTRODUCTORY 

i.  The  purpose  of  machinery  is  to  transform  energy  obtained 
directly  or  indirectly  from  natural  sources  into  useful  work  for 
human  needs.  Useful  work  involves  both  motion  and  force,  hence 
the  basis  of  Machine  Design  is  the  laws  that  govern  motion  and 
force. 

The  term  useful  work  carries  with  it  the  idea  of  definite  motion 
and  definite  force,  for  work  itself  is  always  of  a  definite  or  meas- 
urable character.  An  examination  of  any  machine  will  show 
that  its  parts  are  so  put  together  as  to  give  definite  constrained 
motion  suitable  for  the  work  to  be  done.  The  constrainment  of 
motion  is  determined  by  the  moving  parts,  the  stationary  frame 
and  the  nature  of  the  connections  between  them. 

Mechanics  is  the  science  which  treats  of  the  relative  motions 
of  bodies,  solid,  liquid,  or  gaseous,  and  of  the  forces  acting  upon 
them. 

Mechanics  of  Machinery  is  that  portion  of  pure  mechanics 
which  is  involved  in  the  design,  construction,  and  operation  of 
machinery.  It  has  been  noted  that  the  consideration  of  a  ma- 
chine involves  constrained  motion,  hence  that  portion  of  pure  me- 
chanics is  mostly  needed  in  Machine  Design  which  deals  with 
stationary  structures  and  constrained  motion.  While  the  laws 
of  Mechanics  of  Machinery  give  us  the  underlying  principles  on 
which  machine  action  rests,  their  practical  application  brings  in 
many  modifying  conditions. 

Machine  Design  therefore  may  be  defined  as  the  practical  ap- 
plication of  Mechanics  of  Machinery  to  the  design  and  construc- 
tion of  machines. 


2  MACHINE    DESIGN 

A  Mechanism  is  a  combination  of  material  bodies  so  con- 
nected that  motion  of  any  member  involves  definite,  relative, 
constrained  motion  of  the  other  members.  A  mechanism  or  com- 
bination of  mechanisms  which  is  constructed  not  only  for  modify- 
ing motion  but  also  for  the  transmission  of  definite  forces  and  for 
the  performance  of  useful  work  is  called  a  machine.  A  machine 
consists  of  one  or  more  mechanisms;  a  mechanism,  however,  is 
not  necessarily  a  machine.  Many  mechanisms  transmit  no  energy 
except  that  required  to  overcome  their  own  frictional  resistance, 
and  are  used  only  to  modify  motion  as  in  the  case  of  most  engi- 
neering instruments,  watches,  models,  etc. 

A  brief  reflection  will  show  that  the  same  mechanism  will  serve 
for  different  machines  (see  any  treatise  on  Kinematics)  and 
within  limits  the  design  of  the  mechanism  for  a  given  machine 
may  usually  be  carried  out,  so  far  as  motion  is  concerned,  with  lit- 
tle regard  to  the  amount  of  energy  to  be  transmitted.  This,  of 
course,  does  not  apply  to  such  mechanisms  as  centrifugal  gover- 
nors, or  in  general  where  inertia  or  other  kinetic  actions  affect 
constrainment  of  motion.  Except  for  the  limitations  of  such 
cases  as  those  just  noted,  the  design  of  any  machine  may  be  di- 
vided into  two  main  parts : 

(1)  Design  of  the  mechanism  to  give  the  required  motion. 

(2)  Proportioning  of  the  parts  so  that  they  will  carry  the 
necessary  loads  due  to  transmitting  the  energy,  without  undue 
distortion  or  practical  departure  from  the  required  constrained 
motion. 

(i)  The  design  or  selection  of  the  mechanism  for  a  machine 
is  governed  by  the  manner  in  which  the  energy  is  supplied  and 
the  character  of  the  work  to  be  done;  for  energy  may 'be  sup- 
plied in  one  form  of  motion  and  the  work  may  have  to  be  done 
with  quite  a  different  one.  If  mechanisms  already  exist  which 
will  accomplish  the  desired  result  the  problem  is  one  of  selection 
and  arrangement  of  parts.  But  if  a  new  type  of  machine  is  to 
be  built,  or  a  new  mechanism  is  desired,  the  solution  of  the  mo- 
tion problem  borders  on  or  may  indeed  be  of  the  nature  of  inven- 
tion. While  it  is  true  that  in  most  cases  the  mechanism  and  the 
relative  proportions  of  its  parts  can  be  designed  to  suit  the  work 


INTRODUCTION  3 

to  be  done  without  reference  to  the  energy  transmitted,  in  general 
it  is  necessary  to  know  something  about  the  energy  trans- 
mitted before  any  definite  dimensions  of  the  parts  of  the  mechan- 
ism can  be  fixed,  and  frequently  before  the  nature  of  the  mech- 
anism is  determined.  Furthermore,  the  methods  and  available 
facilities  of  construction  control  the  design  to  a  large  extent. 
Thus  in  designing  a  steam  engine  the  size  of  the  cylinder  must 
be  first  fixed  before  the  length  of  crank  and  connecting-rod  can 
be  fixed,  and  in  general  while  the  mechanism  can  be  treated  apart 
from  the  energy  problem*  it  is  necessary  to  keep  the  latter  con- 
stantly in  mind. 

(2)  The  problem  of  proportioning  the  various  parts  of  a  ma- 
chine so  that  they  will  carry  their  loads  without  excessive  or  un- 
due deformation  may  conveniently  be  divided  into  two  parts: 

(a)  Solution  as  a  whole,  of  the  energy  and  force  problem  in 
the  mechanism. 

(b)  Assigning  of  dimensions  to  the  various  parts  based  on 
the  forces  acting  upon  them. 

(a)  When  the  type  and  proportions  of  the  mechanism  have 
been  fixed  the  relative  velocity  of  any  point  in  the  mechanism 
may  be  found.     If  then  the  energy  which  the  mechanism  must 
transmit  is  known,  it  is  possible,  in  general,  to  find  the  forces  act- 
ing at  any  point  since  the  law  of  Conservation  of  Energy  under- 
lies all  machines;   or  the  product  of  velocity  multiplied  by  force 
is  constant  throughout  the  train.     If  the  forces  acting  on   a 
machine  member  and  the  manner  in  which  it  is  connected  are 
known,  these  may  serve  as  a  basis  for  the  assigning  of  definite 
dimensions  to  the  part.     A  fuller  discussion  of  this  important 
principle  is  given  in  Chapter  III. 

(b)  If  the  forces  acting  on  a  machine  member  can  be  deter- 
mined it  would  seem  easy  to  choose  the  material  and  assign  pro- 
portions to  it  based  on  the  laws  of  Mechanics,  and  such  is  the 
case  when  the  stresses  are  simple  and  the  conditions  fully  known. 
Thus  a  machine  member  subjected  to  simple  tension  within 
known  limits,  can  be  intelligently  proportioned  in  this  manner. 
But  in  many  cases  the  forces  acting  are  very  complex,  the  theo- 
retical design  is  not  always  clear,  and  our  knowledge  of  materials 


4  MACHINE    DESIGN 

and  their  laws  is  limited  in  many  respects.  Recourse  must  there- 
fore often  be  made  to  judgment  or  to  empirical  data,  the  result  of 
experience.  Even  when  the  conditions  are  clear,  theoretical  de- 
sign must  always  be  tempered  with  practical  modification  and  by 
constructive  considerations,  etc.  The  logical  method  of  propor- 
tioning machine  elements  where  theory  is  applicable  is,  therefore, 
as  follows: 

(a)  Make  as  close  an  analysis  as  possible  of  all  forces  acting 
and  proportion  parts  according  to  theoretical  principles. 

(b)  Modify   such  design   by   judgment  and  a  consideration 
of  the  practical  production  of  the  part. 

In  the  case  of  details  and  unimportant  parts,  judgment  and  em- 
pirical data  are  commonly  the  best  guides. 

Summing  up  then,  the  logical  steps  in  the  design  of  a  machine 
are  as  follows: 

(I)  Selection  of  the  mechanism. 
(II)   Solution  of  the  energy  and  force  problem. 

(III)  Design  of  the  various  machine  members  so  they  will 
not  unduly  distort  or  break  under  the  loads  carried. 

(IV)  Specification  and  Drawing. 

The  last  step,  Specification  and  Drawing,  is  a  necessary  and 
important  adjunct  to  the  process  of  design;  it  is  a  powerful  aid 
to  the  designer's  mental  process  and  is  the  best  way  of  showing 
the  workman  what  is  to  be  done  to  construct  the  machine  in  ques- 
tion, and  also  of  making  a  record  of  what  has  actually  been  done. 
It  is  not  machine  design  of  itself,  however,  as  machines  may  be 
designed  and  built  without  any  drawings.  It  is,  nevertheless, 
an  indispensable  part  of  the  designer's  equipment.  Very  often 
written  specifications  accompanying  the  drawings  are  not  only 
useful  but  necessary.  In  fact  the  highest  skill  on  the  part  of  the 
designer  is  often  needed  to  clearly  and  fully  specify  in  writing 
just  what  is  to  be  done,  as  the  writing  of  specifications  presup- 
poses the  most  intimate  knowledge  of  theory  of  design,  and  selec- 
tion of  materials. 

From  the  foregoing  it  is  seen  that  the  part  of  Machine  Design 
included  in  Mechanism  can  be  and  generally  is  for  convenience 


INTRODUCTION  5 

taught  as  a  separate  subject,  and  the  student  is  expected  to  have 
a  knowledge  of  Mechanism,  Mechanical  Drawing,  Mechanics  of 
Engineering,  and  Materials  of  Engineering  as  a  preparation  for 
the  work  contained  in  this  book.  The  chapters  that  follow  deal 
therefore  with  the  solution  of  the  Energy  and  Force  Problem, 
and  the  Design  of  Machine  Elements. 


CHAPTER  II 
THE  ENERGY  AND  FORCE  PROBLEM 

2.  From  the  law  of  Conservation  of  Energy  it  is  known  that 
energy  can  be  transformed  or  dissipated  but  not  destroyed. 
Therefore  all  the  energy  supplied  to  any  machine  must  be  ex- 
pended as  either  useful  or  lost  work.  Since  frictional  resist- 
ances, and  frequently  other  losses,  occur  in  all  machines,  the 
useful  work  done  must  always  be  less  than  the  energy  received. 
The  useful  work  delivered  divided  by  the  energy  received  is 
called  the  efficiency  of  the  machine.  This  expression  is  differ- 
ent for  different  machines  and  is  evidently  a  fraction  or  less  than 
unity.  In  the  discussion  which  follows  in  this  chapter,  frictional 
losses  are  neglected,  unless  otherwise  stated. 

A  Kinematic  Cycle  is  made  by  a  machine  when  its  moving 
parts  start  from  any  given  set  of  simultaneous  positions,  pass 
through  all  positions  possible  for  them  to  occupy,  and  ultimately 
return  to  their  original  positions. 

The  energy  received  by  a  machine  during  a  kinematic  cycle 
may  or  may  not  be  equal  to  the  work  done  plus  frictional  losses. 
Thus  the  energy  supplied  during  a  number  of  cycles  may  be 
stored  in  some  heavy  moving  part  and  then  be  given  out  during 
some  succeeding  kinematic  cycle,  as  in  the  case  of  a  punching 
machine  with  a  heavy  flywheel. 

An  Energy  Cycle  is  made  by  a  machine  when  its  moving 
parts  start  from  any  given  set  of  simultaneous  energy  conditions, 
pass  through  a  series  of  energy  changes,  and  ultimately  return 
to  their  original  energy  conditions. 

Thus  the  complete  mechanism  of  a  four-stroke  gas  engine 
makes  one  kinematic  cycle  every  two  revolutions  of  the  crank 
shaft.  The  slider-crank  mechanism  of  the  engine  considered 
separately  makes  a  complete  kinematic  cycle  every  revolution  of 
the  crank.  The  engine  makes  one  energy  cycle  every  two  revo- 

6 


THE    ENERGY    AND    FORCE    PROBLEM  7 

lutions  of  the  crank.     If  a  punching  machine  driven  by  a  belt 
and  running  continuously,  punches  a  hole  every  fourth  stroke 
of  the  punch,  it  will  be  making  a  complete  kinematic  cycle 
every  stroke  and  a  complete  energy  cycle  every  four  strokes. 
Therefore,  during  a  kinematic  cycle, 

Energy    received  =  useful    work  +  lost    work  ±  stored    energy. 
And  during  an  energy  cycle, 

Energy  received  =  useful  work  +  lost  work. 

Generally  speaking,  the  useful  work  to  be  done  and  also  the 
character  of  the  source  of. energy  are  known  and  the  problem 
of  design  is,  therefore,  to  select  the  mechanism  which  will  trans- 
form the  motion  of  the  source  of  energy  into  the  required  motion, 
to  determine  the  capacity  of  the  driving  device,  and  to  proportion 
the  machine  members. 

The  proportions  of  any  machine  part  depend,  as  regards 
strength  and  rigidity,  on  the  maximum  force  it  must  carry; 
and  this  maximum  force  may  be  due  to  the  direct  action  of  the 
driving  device,  or  it  may  result  from  the  inertia  effect  of  some 
member  which  has  a  capacity  for  storing  energy,  and  in  such  a  case 
may  be  greatly  in  excess  of  any  direct  force  that  the  driving  de- 
vice may  deliver.  Before  this  maximum  force  can  be  determined 
for  any  member  it  is  therefore  necessary  to  make  a  complete 
solution  of  the  energy  problem  including  the  determination  of 
the  driving  device. 

A  knowledge  of  the  quantity  of  energy  required  to  do  the 
desired  work  during  a  complete  energy  cycle  is  not  always  suf- 
ficient information  upon  which  to  base  the  design  of  the  machine 
or  the  capacity  of  its  driving  device. 

A  machine  may  receive  energy  at  either  a  uniform  or  variable 
rate  and  may  be  called  upon  to  do  work  at  either  a  uniform  or 
variable  rate.  Power  or  rate  of  doing  work  being  the  product 
obtained  by  multiplying  together  simultaneous  values  of  velocity 
and  force,  it  follows  that  in  making  any  energy  transformations 
both  the  force  and  the  velocity  factors  must  be  kept  in  mind. 
While  the  mechanism  chosen  may  transform  the  motion  of  the 
source  of  energy  into  the  desired  motion,  it  may  not  necessarily 


8  MACHINE    DESIGN 

so  modify  the  energy  as  to  give  a  distribution  of  force  at  the 
point  where  work  is  being  done  which  exactly  or  even  approxi- 
mately fulfils  the  required  conditions.  Again,  some  of  the 
moving  machine  parts  may  have  to  be  very  heavy  in  order  to 
carry  the  required  loads,  and  during  one  part  of  the  cycle  they 
may  absorb  energy,  thus  reducing  the  operating  force,  while  at 
another  part  of  the  cycle  they  may  give  up  energy,  thus  increas- 
ing the  operating  force.  Such  a  condition  may  make  an  entirely 
different  distribution  of  the  forces  acting  on  the  members  of  the 
mechanism,  from  that  which  would  occur  were  the  parts  light 
or  the  motion  of  the  machine  very  slow,  and  may  materially 
modify  the  design. 

If  it  is  predetermined  that  some  device  is  to  be  used  for 
storing  energy  when  the  effort  is  in  excess,  and  for  giving  it 
out  when  the  effort  is  deficient,  the  capacity  of  the  driving  de- 
vice need  only  be  such  as  will  supply  during  the  energy  cycle 
an  amount  of  energy*equal  to  the  useful  work  and  lost  work  dur- 
ing that  cycle.  But  in  many  machines  such  devices  are  not 
desirable  and  in  many  others  they  cannot  be  applied. 

Two  such  cases  may  be  noted,  (a)  In  many  machines  under 
continuous  operation,  where  flywheels  are  not  desirable,  it  is 
found  that  if  the  driving  device  is  proportioned  so  as  to  supply 
energy  at  a  uniform  rate  equal  to  the  average  rate  required 
throughout  the  energy  cycle,  the  force  at  the  operating  point  is 
sometimes  greater  and  sometimes  less  than  that  required.  If 
simultaneous  values  of  the  force  and  velocity  at  the  working 
point  are  multiplied  together,  their  product  is  the  rate  at  which 
\vork  will  be  done  at  the  point  considered.  The  maximum 
product  thus  obtained  will  be  the  maximum  rate  at  which  work 
will  be  done  and  also  at  which  energy  must  be  supplied  by  the 
source.  It  is  evident  that  the  capacity  of  the  driving  device 
will  be  greater  in  such  a  case  than  if  based  on  the  average  rate 
of  energy  required  per  energy  cycle.  If  the  driving  device 
under  the  above  conditions  should  be  too  large  or  expensive,  as 
is  liable  to  be  the  case  in  large  work,  recourse  must  be  had  to  a 
different  mechanism  or  to  the  use  of  flywheels  or  other  means 
of  storing  and  redistributing  energy,  (b)  Again,  consider  any 


THE    ENERGY    AND    FORCE    PROBLEM  9 

hoisting  mechanism.  Not  only  must  the  driving  device  supply 
during  the  cycle  of  operations  (the  raising  of  the  load)  energy 
equal  to  the  work  done,  but  it  must  also  be  able  to  start  and 
sustain  the  load  at  any  point.  It  is  evident  that  in  such  cases 
the  torque  of  the  driving  device  on  the  hoisting  drum,  must  be 
at  least  equal  to  that  of  the  load,  and  if  the  torque  of  the  driving 
device  should  be  variable,  its  minimum  torque  must  be  equal 
to  that  of  the  load  when  referred  to  the  same  shaft.*  If  this 
minimum  torque  should  be  small  compared  to  the  maximum, 
the  driving  device  chosen  might  have  to  be  excessively  large 
and  this  condition  might  preclude  the  use  of  the  driving  device 
first  selected. 

In  any  of  these  cases,  after  the  form  and  capacity  of  the 
driving  device  have  been  determined,  the  maximum  force  that 
may  come  on  any  member  may  also  be  determined. 

It  is  to  be  noted  that  the  choice  of  mechanism  and  the  capacity 
of  the  driving  device  are  governed  largely  by  the  relative  manner 
in  which  energy  is  to  be  received  and  work  done,  and.  it  may  be 
well  to  enumerate  the  combinations  that  can  occur,  before  apply- 
ing the  above  principles  to  the  discussion  of  illustrative  problems. 

In  any  machine  under  continuous  operation  energy  may  be 
received  and  work  may  be  done  in  one  of  the  following  ways: 

(a)  Energy  may  be  received  at  a  constant  rate  and  work  be 
done  at  a  constant  rate. 

(b)  Energy  may  be  received  at  a  constant  rate  and  work  be 
done  at  a  variable  rate. 

(c)  Energy  may  be  received  at  a  variable  rate  and  work  be 
done  at  a  constant  rate. 

(d)  Energy  may  be  received  at  a  variable  rate  and  work  be 
done  at  a  variable  rate. 

3.  Case  (a).  As  an  example  of  this  case,  where  energy  is 
received  at  a  constant  rate  and  work  done  at  a  constant  rate, 
consider  a  steam  turbine  running  a  centrifugal  pump  raising 
water  to  a  fixed  level.  Evidently  the  rate  at  which  energy  is 

*  In  certain  hoisting  devices  friction  is  utilized  to  sustain  the  load  or  prevent 
overhauling;  this  statement  does  not  apply  broadly  to  such  cases. 


10  MACHINE    DESIGN 

supplied  must  just  equal  the  rate  at  which  work  is  done  plus  frio 
tional  and  other  losses,  for  any  given  period,  and  the  capacity  of 
the  turbine  is  very  easily  determined. 

4.  Case  (b).  As  an  example  of  this  case  (energy  received  at 
a  constant  rate  and  work  done  at  a  variable  rate)  consider  the 
case  of  a  machine  for  punching  holes  in  boiler  plate.  Here  the 
driving  belt  can  supply  energy  at  a  constant  rate  while  the 
useful  work,  which  is  of  considerable  magnitude,  is  delivered  in- 
termittently. If  the  driving  belt  were  designed  with  sufficient 
capacity  to  force  the  punch  through  the  plate  by  direct  pull  it 
would  have  to  be  very  large.  The  machine  runs  idly  a  large 
portion  of  the  time,  while  the  plate  is  being  shifted,  and  in  a 
machine  of  this  kind  a  device  for  storing  energy,  such  as  a  fly- 
wheel, can  be  used  to  advantage.  The  total  capacity  of  the 
driving  belt  need  only  be  sufficient  to  supply,  during  the  energy 
cycle,  an  amount  of  energy  equal  to  the  useful  work  plus  the 
lost  work.  When  a  hole  is  punched  the  velocity  of  the  wheel  is 
reduced,  the  wheel  giving  up  stored  energy.  During  the  time  that 
the  machine  is  running  idly  the  belt  can  store  up  energy  in  the 
flywheel  by  bringing  its  velocity  up  to  normal.  The  maximum 
force  that  may  be  transmitted  by  the  machine  members  will  be 
based  on  the  maximum  force  at  the  tool  and  will  be  transmitted 
only  by  the  members  that  lie  between  the  tool  and  the  flywheel. 

As  a  second  example  of  these  conditions,  take  the  design  of  a 
small  shaping  machine.  Here  the  useful  work  is  done  during  the 
forward  stroke  of  the  ram.  During  the  return  stroke  frictional 
resistances  only  are  to  be  overcome.  The  resistance  of  the  cut 
during  the  forward  stroke  is  uniform  and  the  speed  of  cutting  is 
limited  by  the  character  of  the  metal  to  be  cut.  During  the  re- 
turn stroke,  however,  the  velocity  may  be  greatly  increased,  the 
limiting  velocity  depending  on  the  mass  of  the  moving  parts,  as 
these  should  be  brought  to  rest  at  the  end  of  the  stroke  without 
shock.  The  machine  is  driven  by  a  belt  which  can  supply  energy 
at  a  uniform  rate  and,  as  noted  above,  the  work  is  done  at  a 
variable  rate. 

Numerous  mechanisms  have  been  devised  to  meet  these  condi- 
tions. Suppose  a  mechanism  such  as  shown  in  Fig.  i  has  been 


THE    ENERGY    AND    FORCE    PROBLEM 


II 


selected.  The  maximum  length  of  the  stroke  is  fixed  by  the 
work  to  be  done  and  the  minimum  length  of  stroke  should  be  3 
or  4  inches.  Continuous  rotary  motion  is  imparted  to  the  crank 
a  through  the  gear  b  of  which  it  forms  a  part.  The  gear  b  is  in 
turn  driven  by  the  pinion  c  which  is  rigidly  attached  to  the  shaft 
d.  On  the  other  end  of  d  is  a  stepped  pulley  having  diameters 


FIG.  i. 

D1  D2  D3  Z>4.  On  the  countershaft  overhead  is  a  mating  stepped 
pulley  so  placed  that  when  the  belt  is  on  the  largest  step  of  the 
machine  D1  it  is  also  on  the  smallest  step  of  the  countershaft 
pulley.  The  crank  pin  on  a  is  adjustable  and  can  be  moved  from 
the  outer  position  as  shown  toward  the  centre  of  the  crank,  so 
that  the  vibrator  e  can  be  -made  to  give  the  ram  R  any  length  of 


12  MACHINE    DESIGN 

stroke  from  the  maximum  (20  inches  in  this  example)  to  a 
minimum  of  3  or  4  inches.  The  range  of  velocity  of  the  tool 
for  any  length  of  stroke  must  be  such  that  it  can  be  lowered  to 
the  cutting  velocity  of  hard  cast  iron  or  tool  steel  and  raised  to 
the  economical  cutting  velocity  of  brass.  With  the  pin  in  its 
extreme  outer  position  and  the  belt  on  the  large  step  Dl  the 
speed  of  the  ram  will  be  a  maximum  for  that  position  of  the 
belt.  As  the  crank  is  drawn  toward  the  centre  (the  belt  re- 
maining in  its  original  position)  the  velocity  of  the  ram  is  obvi- 
ously decreased.  If  now  the  belt  is  shifted  to  a  smaller  step  as 
D2  the  velocity  of  the  ram  will  be  increased,  so  that  at  any  stroke 
variable  speed  may  be  obtained  to  suit  the  metal  to  be  cut.  It  is 
not  desirable  to  use  a  flywheel,  the  inertia  of  the  moving  parts  is 
small,  and  the  problem  is  therefore  to  design  the  driving  belt  and 
proportion  the  machine  members  on  the  basis  of  the  maximum 
pull  which  the  belt  may  be  able  to  exert. 

The  mechanism  transforms  the  uniform  rotary  motion  of  the 
line  shaft  into  the  required  reciprocating  motion.  Consider  the 
crank  pin  at  its  extreme  outward  position  and  the  belt  on  D{. 
The  velocity  diagram  for  full  forward  stroke  under  these  condi- 
tions is  shown,  the  ordinates  of  the  diagram*  representing  the 
velocity  of  the  ram  to  the  scale  that  the  crank  length  represents 
the  uniform  velocity  of  the  crank  pin.  The  diagram  for  the 
backward  stroke  is  not  drawn  since  it  is  not  needed  in  the 
solution  of  the  energy  problem;  but  it  should  in  general  be 
drawn  to  make  sure  that  the  change  in  velocity  at  the  extreme 
ends  of  the  stroke  is  not  excessive.  If  the  belt  supplies  energy 
at  a  constant  rate  the  force  which  it  can  deliver  at  the  tool  will 
vary  inversely  as  its  cutting  velocity.  The  cutting  resistance, 
however,  is  uniform  so  that  while  the  mechanism  produces  the 
desired  transformation  in  motion  it  may  not  give  the  distribution 
of  force  desired. 

To  design  the  driving  device  (or  belt)  for  such  a  mechanism 

*  For  a  full  discussion  of  these  so  called  quick-return  mechanisms  and  the  methods 
of  drawing  velocity  diagrams  see  "  Kinematics  of  Machinery  "  by  John  H.  Barr, 
"  Machine  Design"  by  Smith  and  Marx  and  "Machine  Design,"  Part  I.,  by  F.  R. 
Jones. 


THE    ENERGY    AND    FORCE    PROBLEM  13 

the  operating  conditions  of  the  machine  when  the  belt  has  both 
its  maximum  and  minimum  velocity  must  be  investigated.  The 
maximum  pull  which  a  belt  can  give  is  7\  --  T2  where  Tl  is  the 
allowable  tension  on  the  tight  side  of  the  belt.  (See  Church's 
"  Mechanics,"  page  182.)  The  power*  that  a  belt  can  give  out  is 
therefore  V  (T1  -  T2)  where  V  is  the  velocity  of  the  belt.  Since 
Tl  --  T2  has,  at  all  moderate  belt  speeds,  a  constant  maximum 
value  for  a  given  belt,  the  power  that  a  belt  can  deliver  will  vary 
directly  with  its  velocity.  The  belt  receives  its  energy  from  a 
shaft  running  at  constant  speed  and  when  the  belt  is  on  the 
smallest  step  of  the  countershaft  cone  it  will  also  be  on  the 
largest  step  Dl  of  the  machine  cone  and  will  in  consequence  be 
running  at  its  lowest  velocity,  under  which  condition  its  capacity 
for  delivering  energy  is  a  minimum. 

The  maximum  power  required  for  small  machine  tools  is 
approximately  constant  at  all  speeds;  for  since  the  heating  effect 
which  governs  the  cutting  capacity  of  the  tool  is  proportional  to 
the  work  done,  it  follows  that  as  the  cutting  speed  is  increased 
the  resistance  of  the  cut  must  be  decreased  and  vice  versa,  thus 
keeping  their  product  approximately  constant.  If  then  the  belt  is 
designed  to  have  sufficient  capacity  when  the  ram  is  making  full 
stroke  and  the  belt  is  on  D^  and  hence  at  the  lowest  belt  veloci- 
ty, it  will  have  excess  capacity  when  in  any  other  position.  If  a 
softer  metal  is  to  be  cut  the  velocity  of  the  ram  may  be  increased, 
but  this  can  only  be  done  by  shifting  the  belt  to  a  position  where 
its  velocity  and  hence  its  capacity  will  be  greater. 

As  before  noted,  the  effect  of  moving  the  crank  pin  inward,  the 
belt  remaining  in  the  same  position,  is  to  decrease  the  average 
velocity  of  the  ram.  Therefore  as  the  stroke  is  made  shorter 
the  velocity  of  the  crank,  to  maintain  a  given  cutting  speed,  must 
be  increased  by  shifting  the  belt  to  a  smaller  step  of  the  machine 
cone.  The  other  limiting  condition  is  when  the  ram  is  making 
its  shortest  stroke  and  giving  a  cutting  velocity  high  enough  for 
the  softest  metal  to  be  worked.  The  belt  should  then  be  on  the 
smallest  diameter  Z>4,  and  hence  at  its  highest  speed. 

*  A  full  discussion  of  the  power  transmitted  by  belting  is  given  in  chap.  12. 


14  MACHINE    DESIGN 

An  inspection  of  the  velocity  diagram  when  the  ram  is  making 
full  stroke  shows  that  its  velocity  is  a  maximum  when  the  ram 
is  in  mid  position.  Neglecting  friction  and  inertia,  which  here 
are  small,  the  force  exerted  on  the  ram  will  be  a  minimum  where 
the  velocity  of  the  ram  is  a  maximum  at  any  given  belt  velocity, 
because,  for  a  given  belt  pull  since  no  flywheel  is  used,  force  at  belt 
X  velocity  of  belt  =  force  at  tool  X  velocity  of  tool.  If,  there- 
fore, with  the  ram  making  full  stroke,  the  capacity  of  the  belt 
when  running  on  Dl  is  made  great  enough  to  give  a  force  at  mid 
position  of  the  ram  equal  to  the  required  cutting  force,  it  will 
have  excess  capacity  at  any  other  position;  and  if  this  condition 
does  not  give  too  large  a  belt  the  driving  device  will  be  satisfactory. 
The  maximum  force  that  any  member  may  have  to  sustain  will 
be  based  on  the  maximum  torque  of  the  belt,  which  will  occur 
when  it  is  running  on  D^\  for  since  the  inertia  forces  are  small 
this  torque  will  be  transmitted  directly  to  the  members,  and  the 
resulting  stresses  may  be  easily  computed. 

Example : 

Let  the  greatest  resistance  of  cut     =     800    Ibs. 

"      "  maximum  stroke  of  ram       =     20      inches. 

"     "  minimum  stroke  of  ram       =     4        inches. 

"     "  maximum  length  of  crank    =     6>£       " 

"      "  minimum     "         "    "         =     i#       " 

"      "  max.  cutting  speed  on  shortest  stroke  and  highest 

belt  speed  =  60  ft.  per  min. 
"     "  max.  cutting  speed   on  full   stroke   and-  lowest  belt 

speed  =  25  ft.  per  min. 
Then  in  general, 

linear  velocity  of  crank       max.  linear  velocity  of  ram 
length  of  crank  max.  ordinate  of  diagram* 

Hence  in  this  example  when  the  ram  is  making  full  stroke  at 
lowest  speed, 

*  In  the  mechanism  here  chosen  the  position  of  the  ram  for  maximum  velocity 
can  be  located  by  inspection  and  the  value  of  the  velocity  determined  without 
drawing  the  complete  diagram.  In  general,  however,  the  diagram  must  be  drawn 
in  order  to  locate  the  maximum  ordinate. 


THE    ENERGY    AND    FORCE    PROBLEM  15 

25'  X  6J 
Linear  vel.  of  crank  = =    23.5  ft.  per  min. 

. '.  R.P.M.  of  crank  =    23'5  X  I2    =  6.0. 
2  X  TT  x  6J 

In  a  similar  way  when  the  ram  is  making  the  shortest  stroke 
at  highest  speed, 

Linear  velocity  of  crank  =  42. 5  ft.  per  min. 

Therefore,  R.P.M.  of  crank  =   42'5  X  ",  -  54.1. 

2  X  *  X  ii 

Let  the  gear  ratio  be  8  to  i.  Then  the  minimum  and  maximum 
R.P.M.  of  shaft  ^  =  55.2  and  432.8  respectively.  A  14"  pulley 
is  a  convenient  diameter  for  Z>t. 

,     .        c  i    i  14  X  TT  X  ^.2 

.  * .  velocity  of  belt  on  low  speed  =  — 

=  204  ft.  per  min. 

If  the  efficiency  of  the  machine  be  85  per  cent,  the  maximum 
rate  of  doing  work  at  this  position  of  belt  is  the  cutting  resistance 
multiplied  by  the  maximum  velocity  of  the  ram,  divided  by  the 

800 
efficiency,  or  -  -  X  25  =  23,500  ft.  Ibs.  per  minute. 

•  °5 

.*.  effective  pull  at  belt  =  -^ —  =115  Ibs.  approximately. 

The  effective  pull  of  single-ply  belt  per  inch  of  width  may  be 
taken  at  40  to  45  Ibs. 

.'.  width  of  belt  =  —  =  2%"  nearly. 
45 

If  the  cone  pulleys  on  machine  and  countershaft  are  alike,  as  is 
the  usual  case  in  metal-working  tools,  then 


A  _      JMax.  R.P.M. 
A  "      *Min.    R.P.M. 


of  Machine  Cone 


of  Machine  Cone 


.    7)     _  j)      /Min.    R.P.M.  of  Machine  Cone 
1  ^Max.  R.P.M.  of  Machine  Cone 

mce,  in  the  ex 
nearly. 


and  hence,  in  the  example  if  A  =  r4>  A  =  X4  \j  -    —•  =  5" 

432.8 


1 6  MACHINE    DESIGN 

The  maximum  force  that  may  be  applied  to  any  member 
will  be  based  on  the  maximum  torque  of  the  driving  belt,  which 
occurs  when  the  belt  is  on  D^  the  largest  step  of  the  machine 
cone.  The  difference  in  this  respect  between  this  case  and  the 
punching  machine  discussed  above  should  be  noted,  for,  while 
the  driving  mechanisms  of  both  can  deliver  energy  at  a  uniform 
rate  and  while  both  do  work  at  a  variable  rate,  the  maximum 
load  is  applied  in  entirely  different  ways. 

During  the  complete  energy*  cycle  of  the  machine  the  total 
work  done,  neglecting  friction,  is  equal  to  the  length  of  stroke 
multiplied  by  the  uniform  resistance  of  the  cut,  or 

20 
800  X  —  =  1333  ft.  Ibs.     For  every  cycle  of   the  machine  the 

shaft  d  makes  8  revolutions;  hence  the  amount  of  energy  that  the 
belt  could  deliver  if  work  were  done  uniformly  during  one  cycle 

T  /I     V    •— 

is  8  X  -         -  X  115  =  3370  ft.  Ibs. 

The  capacity  of  the  belt  is  therefore  two  and  one-half  times  as 
great  as  it  would  need  to  be  if  a  device  for  equalizing  the  energy, 
such  as  a  flywheel,  had  been  used.  Where  a  small  machine  is 
belt-driven,  as  in  the  case  discussed,  this  added  first  cost  is  not 
serious.  But  when  the  power  needed  is  great,  or  in  such  cases  as 
direct  driving  by  electric  motor,  the  additional  cost  of  a  driving 
device  so  greatly  in  excess  of  average  requirements  needs  to  be 
carefully  considered.  This,  in  fact,  is  one  of  the  most  important 
elements  to  be  considered  in  fixing  the  size  of  motors  needed 
for  direct-driven  machine  tools,  sometimes  making  it  desirable  to 
introduce  a  flywheel  to  reduce  the  size  of  motor. 

5.  Case  (c).  One  of  the  best  examples  of  Case  (c)  where 
energy  is  received  at  a  variable  rate  and  work  is  performed  at  a 
uniform  rate  is  found  in  the  reciprocating  steam  engine,  and 
since  this  machine  is  of  such  great  importance  to  the  engineer 
it  will  be  discussed  somewhat  in  detail.  Here  the  energy  is 
supplied  in  the  form  of  steam  pressure,  and  after  cutoff  occurs 
and  the  steam  expands  in  the  cylinder  the  pressure  falls  from 

*The  kinematic  and  energy  cycle  are,  in  this  case,  simultaneous. 


THE    ENERGY    AND    FORCE    PROBLEM  17 

the  "initial"  or  boiler  pressure  to  somewhat  above  exhaust  or 
atmospheric  pressure.  The  energy  is  therefore  supplied  at  a 
varying  rate.  But  the  engine  is  required  to  deliver  energy  at 
the  driving  belt  at  a  uniform  rate.  The  mechanism  used  will 


L 


— Stroke 


FIG.  3. 


1 8  .MACHINE    DESIGN 

produce  the  required  transformation  of  the  reciprocating  motion 
of  the  piston  into  the  rotary  motion  of  the  crank  shaft.  But  the 
distribution  of  the  driving  force  in  the  form  of  torque  or  tan- 
gential effort  will  not  be  uniform  but  it  will  be  a  maximum 
somewhere  near  the  position  at  which  the  crank  is  at  right  angles 
to  the  connecting-rod,  and  it  becomes  zero  when  the  crank  is  on 
the  dead  centre.  The  turning  effort  will  therefore  sometimes 
be  greater  and  sometimes  'less  than  the  resisting  effort  of  the 
driving  belt  and  the  machine  will  stop  unless  a  redistributing 
device,  such  as  a  flywheel,  is  used.  The  reciprocating  parts, 
such  as  the  piston  and  crosshead,  and  also  the  connecting-rod, 
are  heavy  and  their  maximum  velocity  is  considerable;  hence 
the  forces  due  to  their  inertia  cannot  be  neglected. 

Referring  to  Fig.  2  (a),  the  crank  a  is  required  to  rotate  around 
the  center  O  with  uniform  velocity  and  to  give  a  uniform  force 
at  the  driving  belt.  The  moment  at  the  driving  belt  is  equal  to 
the  average  moment  at  the  crank  pin,  hence  the  equivalent 
uniform  force  at  the  crank  pin  may  be  derived  from  that  at  the 
belt.  This  required  driving  force  at  the  crank  pin  may  be 
plotted  radially  from  the  crank  circle  as  a  base,  forming  a  polar 
diagram  of  the  required  force  at  the  pin,  as  shown 'by  circle  S. 
The  crosshead  C  moves  at  a  varying  rate  of  speed.  If  the 
velocity  of  the  crank  pin  be  represented  by  the  length  of  the 
crank,  the  intercept  Op  made  by  the  connecting-rod  on  the  ver- 
tical through  O  will  represent  the  simultaneous  velocity  of 
the  crosshead  to  the  same  scale.  These  intercepts  may  be  plot- 
ted at  the  corresponding  positions  of  the  crosshead,  thus  out- 
lining the  curve  whose  ordinates  represent  the  velocity  of  the 
crosshead  at  any  point. 

The  forces  acting  upon  the  piston  and  which  must  be  trans- 
mitted to  the  crank  are, 

(1)  The  steam  pressure  which  is  represented  at  any  point  by 
the  ordinates  of  the  curve  T,  Fig.  2  (b). 

(2)  The  back  pressure*  on  the  other  side  of  the  piston,  act- 

*  This  generally  amounts  to  2  or  3  pounds  per  sq.  in.  above  atmospheric  pressure 
in  non-condensing  engines. 


I 

THE    ENERGY    AND    FORCE    PROBLEM  ig 

ing  against  the  steam  pressure,  and  represented  by  the  exhaust 
pressure  line  2  z  and  the  compression  curve  U. 

(3)  The  inertia  forces  due  to  accelerating  and  retarding  the 
heavy  reciprocating  parts. 

During  the  first  part  of  the  stroke  these  inertia  forces  tend  to 
reduce  the  effective  pressure  transmitted  to  the  crank  pin,  and 
during  the  latter  part  they  increase  the  effective  force  on  the  rod. 
They  can  be  represented  graphically  by  such  a  curve  as  V.  The 
first  two  curves  can  be  found  by  the  well-known  methods  of 
drawing  indicator  cards,  and  the  third  can  be  found  either  by 
mathematical  deduction  or  by  graphic  methods*  based  on  the 
velocity  diagram.  It  is  believed  that  the  analytical  method  is 
the  most  satisfactory,  and  such  a  method  is  presented  in  a  suc- 
ceeding article. 

If  the  acceleration  is  known  the  force  necessary  to  produce  the 
acceleration  is  also  known  since  accelerating  force  =  mass  X  ac- 
celeration, and  the  force  at  any  point  (reduced  to  pounds  per 
sq.  in.  of  piston)  may  be  plotted  as  shown  by  curve  V,  Fig.  2  (b). 
When  the  reciprocating  parts  reach  their  maximum  velocity 
their  acceleration  is  zero,  hence  the  curve  of  acceleration  forces 
crosses  the  axis  at  a  point  g  corresponding  to  the  point  of  maxi- 
mum velocity.  This  point  is  very  nearly  at  the  position  where 
the  crank  and  the  connecting-rod  are  at  right  angles  and  the 
error  introduced  by  assuming  this  to  be  so  is  small  with  ordinary 
ratios  of  crank  to  connecting-rod  length.  Beyond  g  the  recipro- 
cating parts  are  retarded,  hence  the  inertia  forces  increase  the 
effective  crank-pin  pressure  from  that  point  on.  The  compression 
curve  (U)  tends  to  decrease  the  effective  pressure  on  the  piston 
and  hence  its  ordinates  must  be  subtracted  from  the  forward 
pressure.  The  algebraic  sum  of  the  curves  T,  U,  and  V  will  give 
a  resultant  pressure  curve  W,  Fig.  2  (c) ,  whose  ordinates  at  any 
point  represent  the  effective  pressure  acting  on  the  piston  rod  at 
that  point.  This  effective  pressure  is  transmitted  to  the  crank 
by  the  connecting-rod  b.  The  pressure  of  the  rod  against  the 
crank  pin  may  be  resolved  into  two  components,  one  tangential 

*  For  a  full  discussion  of  this  matter  see  "  Kinematics  of  Machinery  "  by  J.  H. 
Barr,  page  71,  paragraph  42. 


20  MACHINE    DESIGN 

to  the  crank  circle  and  tending  to  produce  rotative  motion,  and 
one  radial  along  the  crank  tending  to  produce  compression  or 
tension  in  the  crank  and  friction  in  the  main  bearing.  Only  the 
tangential  force  can  do  useful  work.  If  friction  be  neglected 
the  rate  at  which  work  is  done  by  this  force  at  the  crank  must 
equal  the  rate  at  which  work  is  being  done  at  the  piston.  Now 
the  curves  R  and  W,  Fig.  2  (a)  and  2  (c)  respectively,  give  the  simul- 
taneous values  of  force  and  velocity  at  every  point  of  the  stroke. 
If  such  simultaneous  values  be  multiplied  together  and  divided 
by  the  uniform  velocity  of  the  crank  (all  in  the  proper  units) 
the  quotient  is  the  tangential  force  at  the  pin,  and  this  may  be 
plotted  radially  on  the  crank  circle  as  a  base,  thus  giving  what  is 
called  a  radial  crank-effort  diagram,  Fig.  2  (c),  Curve  X. 

These  values  of  the  tangential  force  can  be  found  more  easily 
graphically.  It  will  be  remembered  that  the  ordinates  of  the 
velocity  diagram  (R)  ,  as  drawn  in  Fig.  2  (a)  ,  represent  the  velo- 
city of  the  crosshead  to  the  same  scale  as  the  length  of  the  crank 
represents  the  velocity  of  the  crank  pin.  In  Fig.  2  (c)  ,  the  connect- 
ing-rod extended,  if  necessary,  cuts  the  perpendicular  through 
O  in  the  point  h.  Therefore  O  h  =  velocity  of  crosshead  when 
O  /  =  velocity  of  crank  pin.  Neglecting  friction,  the  rate  of 
work  at  the  crank  pin  is  equal  to  the  rate  of  work  at  the  cross- 
head,  hence  the  velocity  of  the  crank  pin  multiplied  by  the  force 
at  the  crank  pin  is  equal  to  the  velocity  of  the  crosshead  multi- 
plied by  the  force  at  the  crosshead,  or  the  tangential  force  xOy 
=  elflxOh. 


,.  }f 
tangential  force  = 


Lay  off  O  i  =  ejl  and  draw  i  k  parallel  to  b.     Then,  -^-r  =  —  . 

OixOh      ejtXOh 
Therefore,  O  k  =  --  —  r—  -L-^~.  --  =  tangential  force. 

Therefore  O  k  may  be  laid  off  radially  from  j  as  an  ordinate  of 
the  required  curve  as  /  k'.  The  construction  for  the  return 
stroke  is  performed  in  a  similar  manner. 

It  will  be  noted  that  the  distribution  of  force  as  represented 
by  this  diagram  is  less  uniform  than  the  original  curve  of  press- 


THE    ENERGY    AND    FORCE    PROBLEM  21 

ure  at  the  crosshead.  By  the  conditions  of  the  problem,  how- 
ever, the  mechanism  must  produce  a  uniform  turning  effort  at 
the  driving  belt  or  such  as  would  be  given  by  a  crank-effort  dia- 
gram like  S,  Fig.  2  (a).  A  flywheel  must  therefore  be  used  to 
store  energy  when  the  crank  effort  is  in  excess  and  to  give  out 
energy  when  the  crank  effort  is  deficient.  Fig.  2  (d)  shows  the 
crank-effort  diagram  rectified  with  rectangular  ordinates  equal 
to  the  polar  ordinates  of  curve  X.  The  base  YY  is  equal  to  the 
circumference  of  the  crank  circle  and  the  ordinates  of  the  line 
/  m  are  equal  to  the  ordinates  of  the  required  uniform  crank- 
effort  curve  S.  Since  the  abscissas  represent  space  and  the  ordi- 
nates represent  force,  the  areas  /,  K,  J,  Il}  K^  etc.,  represent 
-work.  The  work  represented  by  K+Kl  is  that  which  the  fly- 
wheel must  absorb  and  the  area  represented  by  /  +  7  +  /J  +  /J  that 
which  it  must  give  up  in  one  revolution.  Manifestly  I  +  J  +  It  + 
JL  must  equal  K+K^.  A  full  discussion  of  the  design  of  the 
flywheel  will  be  given  in  a  later  chapter. 

The  maximum  force  that  may  come  upon  the  crosshead  can 
be  seen  from  an  inspection  of  the  force  diagram  W.  It  is  to  be 
noted  in  this  regard  that  if  the  engine  is  designed  for  variable 
cutoff,  an  indicator  diagram  at  late  cutoff  should  be  drawn  for 
the  purpose  of  locating  this  maximum  force,  as  an  earlier  cutoff 
will  not  give  the  maximum  value.  The  method  of  analysis  de- 
veloped above  will  enable  the  designer  to  determine  the  maxi- 
mum straining  action  on  any  member  of  the  mechanism. 

The  graphical  method  of  finding  the  inertia  curves,  while  con- 
venient, are  open  to  criticism  on  account  of  their  inaccuracy  be- 
cause the  tangents  or  sub-normals  to  the  curve,  on  which  these 
graphic  methods  depend,  are  difficult  to  construct  with  accu- 
racy and  are  at  some  points  indeterminate.  In  general,  there- 
fore, it  is  thought  that  the  following  method  or  some  similar 
one  is  more  satisfactory. 

Referring  to  Fig.  3  (page  17), 

Let  a  =  acceleration  at  any  point, 
"   R  =  length  of  crank  in  feet, 
"    L  =     "       "  connecting-rod  in  feet, 


22  MACHINE    DESIGN 

Let  N  =  Rev.  per  min., 

"  0  and  <p  =  angles  made  with  centre  line  by  the  crank  and 
connecting-rod  respectively  at  any  position  measured  from  the 
crank  position  O  r, 

Let  k  =  distance  from  centre  of  crank  shaft  to  mid  position  of 
crosshead, 

Let  x  =  displacement  of  crosshead  from  mid  position, 


"  v  =  velocity  of  crosshead  at  any  point  x, 

"  t   =  time  elapsed  corresponding  to  v. 

"  &  =  angular  velocity  in  radians  per  second, 

Then  x  +  k  =  O  B  +  B  C  =  R.  cos  o  +  L  cos  ?, 


But  L  cos  <?  =  VL2—R2sm20  =  R      —:  —  sin2  9 


.'.  x  +  k  =  R  (cos0  +  Vn2  —  sirfo).  .  .  ..,  ,  (i) 
Expanding  the  radical  by  the  binomial  theorem  and  omitting  all 
terms  beyond  the  second  (which  can  be  done  without  appreciable 
error  with  the  limiting  proportions  ordinarily  used)  equation  (i) 
becomes, 

r  /         sin2  fl\  -i 

x  +  k  =  R  I  cos  o  +  (  n  —  -  )          .....       (2) 

Now  #  =  the  distance  moved  through  by  the  crosshead,  from  mid 

dx 
stroke  and  velocity  at   x  =  —  ;    and  therefore   differentiating 

(2)  with  reference  to  / 

dx  /'.  sin2#W0 

(3) 


The  acceleration  = 

dv      d?x  cos2#\     de 


do  2-N 

but  —  =  angular  velocity  in  radians  per  sec.   =  , 

.....     (5) 


THE    ENERGY    AND    FORCE    PROBLEM 


23 


which  is  the  general  expression  for  acceleration  of  the  recipro- 
cating parts. 

If  the  weights  of  parts  be  called  W,  from  Mechanics  it  is 
known  that  the  force  necessary  to  produce  an  acceleration  (a)  is 

W 

P  =  — a  where  g  =  32.2  in  English  units;  therefore 

o 

Wn(27tN2\(  COS20\ 

" R  ^-— —  J\cos  o  -i — 1   where  R  is  in  ft.     .     (6) 

or  reducing, 

WrN2,  cos20\ 

"  = (  cos  o  4-  -        —I  where  r  is  in  inches.       .      (j) 

When  the  solution  of  the  above  expression  gives  a  negative 
result  the  force  of  inertia  is  acting  away  from  the  crank  and 
when  positive,  toward  the  crank.  It  is  also  to  be  noted  that  the 

W      (2-N2\ 
expression — R  ^— — J  is  the  centrifugal   force   of   a  weight 

equal  to  that  of  the  reciprocating  parts  concentrated  at  the  crank 
pin  since  centrifugal  force  in  general  is  equal  to  —      — . 

o 

By  means  of  equation  (j)  all  points  on  the  acceleration  curve 
could  be  found  and  plotted.  In  general,  however,  the  exact 
characteristics  of  the  curve  are  not  essential  and  it  is -sufficient  to 
make  the  three  most  simple  solutions  as  follows,  and  a  curve 
drawn  through  the  three  points  thus  located  is  sufficiently  accu- 
rate for  all  ordinary  purposes.  In  cases  of  extremely  high  speed 
with  small  ratios  of  connecting-rod  to  crank  a  more  accurate  de- 
termination of  the  curve  may  be  desired. 

/  T      \ 

*      (8) 


When  o  =  00°  or  270°*  P  =  ( — - — )  (-)  .      .    (10) 

V  35200  /  W 


*  The  piston  is  not  at  half  stroke. 


24  MACHINE    DESIGN 

If  the  inertia  forces  are  to  be  combined  with  the  steam  press- 
ures, as  shown  graphically  in  Fig.  2  (6) ,  they  must  be  reduced  to 
pounds  per  square  inch  of  piston  to  give  correct  diagrams. 

An  example  may  serve  to  make  these  points  clearer.  Let  it 
be  required  to  design  a  steam  engine  to  deliver  150  H.P.  with 
the  following  data: 

Steam  pressure  =  90  Ibs.  gauge.     Cutoff  at  }i  stroke. 

Ratio  of  crank  to  connecting-rod  =  i  to  5. 

Piston  speed  =  strokes  per  minute  multiplied  by  length  of 
stroke  =  640  ft. 

Here  something  must  be  known  about  the  size  of  cylinder 
necessary,  before  definite  dimensions  are  assigned  to  the  various 
members.  Let  a  theoretical  indicator  card  be  drawn  as  in  Fig. 
2  (b),  neglecting  for  the  present  the  inertia  curve  V  since  this 
only  tends  to  redistribute  the  energy  and  does  not  affect  its 
quantity.  The  distance  z  z  represents  the  piston  travel  and  the 
ordinates  of  the  curve  T  represent  piston  pressures;  therefore 
the  area  between  z  z  and  the  curve  T  represents  the  work  done 
by  the  steam  pressure  during  the  stroke.  In  a  similar  way  the 
area  under  curve  U  represents  the  work  of  compression  due  to 
back  pressure.  The  difference  of  these  areas  is  the  net  work 
done  per  stroke  of  piston  and  the  mean  ordinate  corresponding 
to  this  area  represents  to  the  proper  scale  the  average  pressure 
per  sq.  inch  on  the  piston  during  stroke.  In  the  case  given 
z  z  =  2ff.  Area  under  T  minus  area  under  U  =  1.75  sq.  in.  There- 
fore mean  ordinate  =  =  -875"-  The  scale  of  pressures 

taken  is  i"  =  70  Ibs.     Therefore  mean  pressure  during  stroke 
=  70  X  .875  =  62  Ibs. 

Let  A  =  area  of  piston. 

P  =  mean  effective  pressure  per  sq.  in. 
L  =  length  of  stroke  in  feet. 
N  =  number  of  revolutions  per  minute. 
H.P.  =  horse  power  required. 

Then  H.P.  =  2  PLAN    ^^  p  N  X  L  and  H.P.  are  known. 
33000 


THE    ENERGY    AND    FORCE    PROBLEM  25 


H  .  P. 

Whence  A  =  -  -I3,  square  inches, 


or  a  diameter  of  cylinder  of  13  inches. 

If  the  stroke  be  taken  at  about  twice  the  diameter  of  the  cyl- 
inder, or  say  24  inches,  the  proportions  will  be  good. 

Hence  since  iL  X  N  =  640,  N  =  i6o  R.P.M.  The  mechan- 
ism can  now  be  laid  out  to  scale.  This  has  been  done  in  Fig.  2 
(a  and  c),*  the  space  scale  being  i"  =  i  ft. 

As  before  stated,  the  location  of  the  three  points,  namely, 
where  0  is  respectively  o°,   180°,   and  90°  or  270°  (Fig.  3),  is 
sufficient  to   locate  the  inertia   curve.     In  the  above  example 
W  =  3.5,  n  =  5,  and  N  ••=  160. 
The  general  expression  for  the  inertia  force  is,  for  0=0. 

WrN2f         i\  (         i\ 

P  =  ~  1  1  +  -  J  =  C  (  i  +  -  )  where  C  is  a   constant 

35,200    ^         n'  n' 

3.S  X  12  X  i6o2 

and  here  equal  to  -  -  =  30.  5. 

35.200 

Therefore,  When  o  =  o°,  P  =  30.5  (i  +  -j  =  36.6lbs. 
When  e  =  90°,    P  =  30.5  (-)  =  6.1  Ibs. 

When  o  -  1  80°,  P  =  30.5  (i  -    -)  =  24.4  Ibs. 

These  values  serve  to  locate  the  curve  as  in  Fig.  (2). 

The  resultant  of  T  U  and  V,  curve  W,  Fig.  2  (c),  can  now  be 
drawn  and  the  crank-effort  diagram  X  plotted.  The  crank-effort 
curve  can  be  rectified  as  in  Fig.  2  (d)  and  the  mean  ordinate  Yl 
drawn.  The  area  I  +  J=K  will  be  proportional  to  the  energy 
to  be  absorbed  and  delivered  by  the  flywheel.  One  inch  of  ordi- 
nate here  =  70  Ibs.  per  sq.  in.  of  piston  and  one  inch  of  abscissa 
=  1  ft.  ;  therefore  one  sq.  in.  of  area  =  70  ft.  Ibs. 

The  area  of  ^  =  .5  sq.  in.  and  area  of  piston  =  132  sq.  in. 
Hence,  if  E  =  energy  to  be  absorbed, 

£=..5X70X132=4,620    ft.    Ibs.    on    which    the    de- 
sign of  the  flywheel  can  be  based. 

*  Reduced  in  reproduction  about  one-half. 


26  MACHINE    DESIGN 

The  maximum  pressure  that  can  occur  on  the  piston  is  the 
initial  or  boiler  pressure  as  the  ordinates  of  W  are  at  all  points 
less  than  those  of  T.  Hence,  when  running,  the  parts  will  be 
subjected  to  less  load  than  in  starting  up,  when  full  boiler  press- 
ure may  be  applied  before  inertia  forces  become  noticeable. 

6.  Case  D.  A  good  example  of  energy  supplied  at  a  vary- 
ing rate  and  work  done  at  a  varying  rate  is  found  in  a  direct- 
driven  air  compressor.  Here  the  varying  steam  pressure  in  the 
steam  cylinder  is  opposed  by  a  varying  air  pressure  in  the  air 
cylinder  as  shown  in  Fig.  5  (a).  The  area  of  the  cylinders  are, 
for  simplicity,  assumed  to  be  equal.  The  steam  cylinder  takes 
steam  at  80  Ibs.  pressure  and  the  air  compressor  cylinder  delivers 
air  at  100  Ibs.  pressure.  The  efficiency  of  the  system  shown  is 
taken  at  80  per  cent,  and  hence  the  area  of  the  compressor  card 
is  80  per  cent,  of  the  steam  card.*  If  both  the  pistons  were 
rigidly  attached  to  the  same  rod  it  is  evident  that  the  maximum 
steam  pressure  will  occur  where  the  air  pressure  is  a  minimum. 
If,  however,  each  cylinder  is  independently  connected  to  a  com- 
mon shaft  by  means  of  a  crank  and  connecting-rod  mechanism, 
the  maximum  and  minimum  pressures  of  the  cards  may  be  made 
to  coincide  more  closely  by  placing  the  crank  pins  at  the  proper 
angular  distance  apart.  In  other  words  the  mechanism  may  be 
so  designed  that  energy  will  be  delivered  at  the  working  point 
more  nearly  at  the  rate  required  by  the  work  to  be  done.  The 
loss  by  friction,  etc.,  is  about  20  per  cent.  Part  of  this  is  lost 
on  the  steam  side  and  part  on  the  air-compressor  side.  It  can 
be  assumed,  without  great  error,  that  the  losses  can  be  evenly 
divided  between  the  two  slider-crank  chains  and  also  that  the 
loss  is  at  a  uniform  rate  throughout  the  stroke.  Thus  the  loss 
on  the  steam  side  can  be  represented  by  the  line  a  b,  Fig.  5  (a), 
which  reduces  the  effective  pressure  at  every  point  by  a  fixed 


*  In  the  general  case,  where  the  cylinders  are  of  different  diameter  and  area, 
the  diagrams  which  represent  pounds  per  square  inch  of  piston  area  would  not 
have  a  ratio  equal  to  the  efficiency.  The  mean  effective  pressure  of  the  air  cylinder 
multiplied  by  the  area  of  the  air  cylinder,  divided  by  the  mean  effective  pressure  of 
the  steam  cylinder  multiplied  by  the  area  of  the  steam  cylinder,  would,  in  this  case, 
equal  the  efficiency. 


THE    ENERGY    AND    FORCE    PROBLEM 


27 


amount.  In  a  similar  way  ordinates  to  the  line  c  d  increase  the 
effective  resistance  of  the  air  diagram.  The  area  of  the  diagrams 
modified  in  this  way  will  be  equal  and  all  energy  supplied  will 
be  accounted  for. 


FIG.  5  (b). 


FIG.  5  (c). 


FIG.  5  (d). 

Since  the  moving  parts  of  both  slider-crank  chains  will  be 
heavy,  the  effect  of  inertia  cannot  be  neglected.  In  Fig.  5  (b) 
the  air  and  steam  cards  are  shown  with  the  inertia  curve,  the 


28  MACHINE    DESIGN 

friction  line,  and  the  compression  curves  in  their  correct  relation- 
ship. Fig.  5  (c)  shows  the  resultant  pressure  curves,  the  curve 
of  air  pressures  being  plotted  below  the  base  line  for  conveni- 
ence. The  crank-effort  curve  of  the  steam  cylinder  is  repre- 
sented by  X,  and  the  resisting  crank-effort  curve  of  the  air  cyl- 
inder is  represented  by  Y.  The  cranks  are  here  placed  90° 
apart,  the  steam  crank  being  in  advance,  a  common  arrangement 
in  practice.  It  is  evident,  however,  that  this  is  not  the  most 
advantageous  angle,  for  if  the  point  e  on  the  air  curve  is  made 
to  correspond  with  /  on  the  steam  curve,  Fig.  5  (c),  the  excess 
and  deficiency  of  effort  will  be  still  further  reduced.  This  would 
place  the  cranks  at  45°  apart.  This  is  even  more  clearly  shown 
in  Fig.  5  (d),  on  the  rectified  curve  of  crank  effort.  Here  the 
area  K+K^  is  the  amount  of  energy  to  be  absorbed  and  7  + J  + 
/i-J-7,  the  amount  to  be  given  up  by  the  flywheel  during  one 
revolution.  In  the  steam  slider-crank  mechanism  the  greatest 
pressure  is,  as  before,  that  due  to  the  initial  steam  pressure, 
while  on  the  air  side  it  will  be  that  due  to  the  terminal  air 
pressure. 

6.1.  In  the  four  cases  discussed  above  the  action  of  the  ma- 
chine has  in  all  instances  been  supposed  to  be  continuous,  and  all 
machines  which  operate  continuously  will  belong  to  one  of  these 
classes.  Where  the  action  of  the  machine  is  intermittent  or 
irregular,  these  general  solutions  will  not  always  hold  and  the 
design  of  the  machine  cannot  be  based  on  the  energy  given  or 
received,  but  will  depend  on  the  maximum  force  or  maximum 
torque  or,  in  other  words,  on  the  mechanical  advantage  which 
the  motor  must  possess.  Thus  the  motor  on  an  auto  car  has  a 
certain  maximum  capacity  for  delivering  power.  On  a  level 
road  it  can  propel  the  car  at  a  high  rate  of  speed,  the  engine 
making  only  a  few  turns  to  every  revolution  of  the  wheels.  But 
on  a  steep  hill  the  gears  must  be  shifted  so  that  the  engine  has  a 
greater  mechanical  advantage,  and  gives  a  greater  torque  on  the 
axle,  the  engine  making  many  revolutions  to  every  one  of  the 
wheels.  Another  example  of  this  is  the  case  of  hoisting  mech- 
anisms already  discussed  somewhat  (see  article  2).  An  en- 
gine or  a  motor  might  be  capable  of  giving  out  energy  at  a  rate 


THE    ENERGY    AND    FORCE    PROBLEM  29 

equal  to  that  required  to  lift  the  load  in  a  given  time,  and  it 
might  be  able,  running  continuously,  to  raise  the  load  to  the 
required  height.  But  its  ability  to  start  and  sustain  the  load  at 
any  point  will  depend  on  whether  it  has  a  mechanical  advantage 
at  that  point  and  not  on  its  capacity.  Where  the  torque  of  the 
load  is  constantly  changing,  as  in  deep  mine  hoisting,  the  design 
of  the  hoisting  devices  becomes  quite  complicated  and  is  beyond 
the  scope  of  the  present  treatise.  It  will  be  noted,  however, 
that  in  such  cases  the  minimum  torque  of  the  motor  or  engine 
must  always  exceed  the  maximum  torque  of  the  load  when  re- 
ferred to  the  same  shaft.  This  general  principle  must  be  kept 
in  mind  in  designing  hoisting  devices  and  similar  machines 
which  act  intermittently  and  slowly,  or  where  redistributing 
devices  are  undesirable  or  impossible. 

6.2.  Redistribution  of  Energy  and  Inertia  Effects.      Devices 
for  storing  and  redistributing  energy  are  very  common  in  transmis- 


• 
f 

al 

e 

iin 

MWW 

I                  A         !                I              A 

' 

|                       B    mm  ^                                                     | 

FIG.  $  (e). 

sion  systems.  Thus,  in  hydraulic  distribution,  the  excess  supply 
of  power  is  stored  in  an  accumulator,  and  given  out  again  when 
the  supply  is  deficient.  In  electrical  distribution  a  storage  battery 
is  sometimes  used  for  the  same  purpose.  In  transmission  of  power 
by  compressed  air  a  large  reservoir  is  sometimes  employed  as  a 
store-house  of  energy.  In  the  case  of  a  single  machine,  the  re- 
distribution is  effected  by  compressing  a  gas,  by  using  a  spring,  or 
by  accelerating  and  retarding  some  heavy  moving  part.  Thus 
in  the  steam  engine  the  piston  compresses  steam  in  the  clearance 
space  at  the  end  of  its  stroke,  and  the  energy  so  absorbed  is  re- 
turned to  it  during  the  next  stroke.  Again,  when  the  energy 
supplied  by  the  steam  is  in  excess  of  the  effort  required,  the  fly- 
wheel absorbs  the  excess  and  thereby  has  its  velocity  (and  hence 


30  MACHINE    DESIGN 

its  kinetic  energy)  increased.     When  the  effort  is  in  excess,  the 
wheel  gives  up  the  stored  energy  at  the  expense  of  its  velocity. 

It  does  not  necessarily  follow,  however,  that  all  heavy  moving 
parts  simply  redistribute  the  absorbed  energy  as  useful  work, 
as  the  action  may  be  a  positive  source  of  loss.  In  Fig.  5  (e)  let 
A  be  the  platen  of  a  large  planing  machine,  and  suppose  it  to  be 
making  its  return  stroke,  moving  from  left  to  right.  The  force 
just  necessary  to  slowly  move  the  platen  may  be  represented  by 
the  vertical  ordinates  of  the  diagram  abed.  Suppose  now, 
that  a  greater  force  is  applied,  in  order  to  hasten  the  operation, 
so  that  at  the  position  A',  the  platen  has  been  accelerated  till  its 
kinetic  energy  is  equal  to  the  rectangle  e  g  h  c.  Evidently  the 
platen  will  not  stop  at  the  end  of  the  stroke  if  the  actuating  force 
be  removed  at  A',  as  the  work  of  friction  during  the  remainder 
of  the  stroke  is  less  than  the  stored  energy.  If,  therefore,  the 
"return  "belt  is  removed  at;!'  and  the  "  driving  "belt  applied,  the 
latter  will  slip  upon  the  driving  pulley  till  the  excess  of  energy 
is  absorbed  and  dissipated  as  heat.  If  the  point  A'  has  been  prop- 
erly chosen  the  platen  will  just  stop  at  the  end  of  the  stroke  and 
the  energy  absorbed  by  the  belt  will  equal  the  area  /  g  h  b.  If 
a  spring,  5,  were  fitted  to  the  machine,  so  that  the  work  of  corrb 
pression  from  the  position  A'  to  the  end  of  the  stroke  just  equalled 
the  excess  kinetic  energy  of  the  platen,  at  that  position,  the  return 
belt  could  be  thrown  off  at  A',  and  the  platen  would  stop  at  the 
end  of  the  stroke.  The  energy  stored  in  the  spring  would  then 
be  returned  to  the  platen  on  the  forward  stroke.  This  latter 
action  is  identical  with  that  of  compression  in  the  steam-engine 
cylinder,  Fig.  2,  the  energy  under  the  curve  U  being  returned  to 
the  reciprocating  parts  on  the  next  stroke.  It  is  to  be  noted  in 
this  last  case,  that  even  if  the  work  of  compression  is  not  quite 
equal  to  the  energy  to  be  absorbed  during  the  latter  part  of  the 
stroke,  there  is  no  loss  of  energy  (friction  neglected),  as  what  is 
not  absorbed  by  compression  is  absorbed  at  the  crank  pin  in 
useful  effort. 


CHAPTER  III 
STRAINING  ACTIONS  IN  MACHINE   ELEMENTS 

7.  Nature  of  Forces  acting  in  Machines.  From  the  fore- 
going chapter  it  is  clear  that  machine  members  which  transmit 
energy  are  subjected  to  forces  of  a  varying  character  and  inten- 
sity. Since  the  various  parts  of  a  machine  must  be  constrained 
to  move  in  fixed  paths  it  is  important  that  they  should  neither 
break  or  be  distorted  appreciably  under  the  loads  carried;  that 
is,  the  members  must  be  not  only  strong  but  also  stiff.  The  pro- 
portioning of  machine  elements  as  dictated  by  various  methods 
of  loading  is  therefore  most  important,  and  will  be  considered  in 
this  chapter. 

The  forces  acting  on  a  machine  element  may  be  one  or  several 
of  the  following: 

(a)  The  useful  load  due  to  the  energy  transmitted. 

(b)  Forces  due  to  frictional  resistances. 

(c)  The  weight  of  the  part  itself  or  of  other  partSc 

(d)  Inertia  forces  due  to  change  of  velocity. 

(e)  Centrifugal  or  inertia  forces. 

(f)  Forces  due  to  change  of  temperature. 

(g)  Magnetic  attractions,  as  in  electrical  machinery. 

These  forces  or  loads  may  be  applied  to  a  machine  in  several 
ways.  They  may  act  steadily  in  one  direction;  they  may  act  in- 
termittently in  one  direction,  or  they  may 'be  applied  first  in  one 
direction  and  then  in  the  reverse;  they  may  be  applied  gradually, 
or  suddenly  in  the  nature  of  a  shock. 

A  steady  or  dead  load  is  one  which  is  always  applied  steadily 
in  the  same  direction.  A  live  load  is  one  which  is  alternately 
applied  and  removed.  A  suddenly  applied  load  is  one  imposed 
instantaneously  but  without  initial  velocity.  If  the  load  is  ap- 

31 


32  MACHINE    DESIGN 

plied  with  initial  velocity  as  in  the  case  of  a  blow  from  a  falling 
body,  the  member  is  subjected  to  impact. 

8.  Nature  of  Straining  Actions,  Stress,  and  Strain.  Since 
all  materials  of  construction  are  more  or  less  elastic  a  machine 
element  must  change  its  form  to  some  extent  whenever  subjected 
to  a  load.  This  change  of  form  may  be  very  small  and  tempo- 
rary; it  may  be  a  permanent  distortion;  or  if  the  load  applied 
be  heavy  enough  the  element  may  even  be  ruptured.  Such 
change  of  form,  whether  temporary  or  permanent,  is  called  a 
strain.  When  a  machine  member  is  thus  distorted  under  a  load 
certain  molecular  reactions,  equal  and  opposite  to  the  load  applied, 
are  set  up  within  the  material  and  resist  the  deformation.  Stress 
is  the  term  applied  to  this  internal  reaction  and  is  to  be  clearly 
distinguished  from  strain,  stress  being  in  the  nature  of  a  force 
and  strain  being  a  dimension. 

The  character  of  the  straining  action  and  of  the  stress  which 
results  from  a  given  load  depend  upon  the  direction  and  point  of 
application  of  the  load  (or  forces),  and  upon  the  form,  the  posi- 
tion, and  the  arrangement  of  the  supports  of  the  member.  A 
given  load  may  produce  tension,  compression,  shearing,  flexure, 
or  torsion  or  a  combination  of  these.  Of  course  tension  and  com- 
pression cannot  both  exist  at  the  same  time  between  any  pair  of 
molecules.  Flexure  is  a  combination  of  tensile  and  compressive 
stresses  between  different  sets  of  molecules;  or,  as  it  is  often  ex- 
pressed, in  different  fibres*  of  the  same  body.  Torsion  is  a 
special  form  of  shearing  stress.  Owing  to  the  frequent  occur- 
rence of  flexure  and  torsion  it  is  convenient  to  treat  these  as 
elementary  forms  of  stress. 

The  stresses  due  to  tension,  compression,  and  flexure  are  essen- 
tially molecular  actions  normal  to  the  planes  separating  adjacent 
sets  of  interacting  molecules;  that  is,  the  stresses  increase  or  de- 
crease the  distances  between  these  molecules  along  lines  connect- 
ing them. 

The  primary  straining  effect  of  shearing  and  torsional  actions 
is  displacement  of  adjacent  molecules,  between  which  -the  stress 

*  It  should  be  noted  that  the  term  fibre  is  used  in  a  conventional  sense  when 
discussing  homogeneous  metals,  such  as  iron  and  steel. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  33 

acts,  tangentially  to  the  planes  separating  such  molecules.  In 
uniform  shear  the  interacting  molecules  move  or  are  strained  rel- 
atively with  a  rectilinear  translation.  In  torsional  action  the  ad- 
jacent molecules  each  side  of  a  plane  of  stress  have  a  relative 
motion  or  strain  about  an  axis.  A  brief  reflection  will  show  that 
in  reality  only  two  kinds  of  strain  exist,  namely,  elongation  (con- 
traction if  negative)  and  shearing.  In  a  similar  way  only  two 
corresponding  kinds  of  stress  are  met  with,  namely,  normal  or 
direct,  and  tangential  or  shearing.  But  for  convenience  it  is 
much  more  desirable  to  treat  the  special  cases  previously  men- 
tioned, separately  as  elementary  stresses.  (See  Church's 
Mechanics,  page  201.) 

Machine  members  are  often  subjected  to  combinations  of  these 
simple  stresses,  as  flexure  and  torsion.  Such  stresses  are  called 
Compound  Stresses  and  will  be  more  fully  treated  later. 

When  a  load  is  applied  to  a  piece  of  material  the  strain  which 
results  is  a  function  of  the  load  and  of  the  character  of  the  ma- 
terial involved.  In  general  for  a  given  loading  the  deformation 
is  different  for  different  materials  but  constant  in  its  relation  to 
stress  for  any  one  material.  These  relations  have  been  deter- 
mined experimentally  for  all  the  ordinary  materials  used  in  engi- 
neering, and  works  on  mechanics  of  materials  treat  of  the  sub- 
ject fully.  Enough  will  be  inserted  here  to  make  the  discussion 
complete. 

If  a  bar  of  metal  is  tested  under  an  increasing  tensile  load  and 
the  strain  caused  by  each  successive  load  is  accurately  observed  the 
relation  between  stress  and  strain  can  be  shown  graphically  as  at 
O  a  de  Fig.  6;  such  a  diagram  is  called  a  stress-strain  diagram. 

If  axes  O  X  and  O  Y  are  chosen  and  the  stresses  plotted  as 
ordinates  and  strains  as  abscissas,  it  will  be  found  that  up  to  a 
certain  point  as  a,  either  in  tension  or  compression,  the  curve  so 
formed  is  sensibly  a  straight  line;  that  is,  stress  is  proportional 
to  strain.  Further,  if  at  any  point  below  a  the  stress  is  released, 
the  piece  returns  to  its  original  shape.  But  above  a  this  relation 
ceases,  strain  usually  increases*  faster  than  stress,  till  finally 

*  Ordinary  rubber  is  an  exception  to  this  general  rule,  strain  decreasing  as 
stress  increases. 


34 


MACHINE    DESIGN 


rupture  occurs.  If  at  any  point  beyond  a  the  stress  is  released, 
it  is  found  that  the  piece  no  longer  returns  to  its  original  dimen- 
sions but  has  been  permanently  distorted. 

If  at  any  point  on  the  curve  below  a  the  stress  be  divided  by 
the  strain  a  ratio  is  obtained  which  is  constant  for  all  points 
below  a.  This  ratio  is  called  the  modulus  or  coefficient  of 
elasticity.  If,  therefore,  this  modulus  of  elasticity  is  known  for 
a  given  material,  the  strain  corresponding  to  any  given  load  may 
be  calculated,  providing  it  does  not  exceed  the  value  correspond- 
ing to  the  point  a. 

The  point  a  is  called  the  elastic  limit  and  is  well-defined  in 
most  materials.  Cast  iron  has,  however,  no  well-defined  elastic 


limit  and  little  permanent  elongation.  Materials  of  this  kind  are 
said  to  be  brittle. 

If  sufficient  tensile  stress  is  applied  to  a  test  piece  its  elonga- 
tion increases  until  finally  it  " necks  down"  at  its  weakest  point 
and  rupture  occurs.  The  load  per  unit  area  under  which  a  bar 
breaks  is  called  its  ultimate  strength  and  the  corresponding  stress 
or  load  per  unit  area  is  called  the  ultimate  stress.  Similar  phe- 
nomena are  observed  when  a  piece  is  tested  in  compression  or 
torsion,  etc. 

It  is  evident  that  the  working  stress  of  a  machine  member 
must  be  less  than  the  elastic  limit  if  the  piece  is  to  retain  per- 
manency of  form.  The  stress  at  which  a  member  is  designed  to 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  35 

be  operated  is  called  the  working  stress  and  the  ratio  of  the 
ultimate  stress  to  the  working  stress  is  called  the  factor  of  safety. 
It  is  to  be  especially  noted  that  not  only  must  the  working  stress 
in  the  member  be  kept  below  the  value  where  permanent  deforma- 
tion takes  place,  but  also  so  low  that  the  resulting  strain,  whatever 
it  may  be,  shall  be  so  small  as  not  to  destroy  the  proper  alignment 
of  the  piece,  or  cause  unnecessary  friction  through  distortion.  A 
machine  member  may  be  amply  strong  enough  to  carry  the  load 
with  perfect  safety,  and  yet  distort  so  badly  under  the  load  as  to 
render  it  unfit  for  the  service  desired.  Both  strength  and  stiffness 
should  therefore  be  kept  in  mind  in  designing  a  machine  part,  as 
sometimes  one  and  sometimes  the  other  will  dictate  the  form  and 
dimensions  to  be  used.  A  short  discussion  will  now  be  given  of 
the  relations  which  exist  between  load,  stress,  and  strain  for  the 
cases  most  often  met  and  of  their  bearing  on  the  selection  of  the 
form  and  size  of  a  machine  member.  In  this  discussion  it  will  be 
assumed  that  the  load  is  a  dead  load  applied  without  shock,  and 
the  modifying  effect  of  suddenly  applied  and  repeated  loads  will 
be  considered  after  the  fundamental  relations  between  load  and 
stress  are  established. 

9.  Tension.  Let  p  be  the  stress  in  the  section,  P  the  load, 
and  A  the  area  of  cross  section.  The  relation  which  exists 
between  them  in  simple  tension  is 


And  if  E  be  the  coefficient  of  elasticity  and  /  the  length  of  the 
member,  the  total  elongation  A  is  given  by  the  equation 

••  A       "       ......    (B) 

A  E 

A 

The  elongation  per  unit  of  length  or  the  strain  =  —r. 

If,  then,  a  tension  member  is  to  be  designed  to  join  two 
machine  parts,  the  formula  for  strength  dictates  a  piece  of  uni- 
form cross  section  without  regard  to  any  particular  form.  Hence 
the  most  convenient  or  cheapest  form  would  be  used,  avoiding 


36  MACHINE    DESIGN 

thin,  wide  sections  where  concentrated  stress  at  the  edge  might 
cause  undue  strain. 

Suppose  it  is  required  to  hold  the  two  surfaces  within 
certain  limits,  as  is  often  the  case  in  machine  tools  where  accu- 
racy is  desired.  If  the  tension  member  is  long  it  may  yield 
more  than  is  desirable,  though  the  working  stress  may  be  well 
below  the  elastic  limit  and  a  greater  area  may  be  necessary  to 
reduce  A  to  the  desired  value. 

Example.  Let  P  =  20,000  Ibs.,  let  the  allowable  stress  p  = 
10,000  Ibs.,  let  £  =  30,000,000,  let  /  =  4o",  and  let  it  be  required 
to  keep  A  within  .001".  If  the  design  is  based  on  allowable  stress 
alone, 

P       20,000 

A  =  —  =  --  =  2  ov,  aare  inches. 
p       10,000 

PI  20,000  X  40 

But  for  A  =  .001,  A  =  —  -  =  —  --  =  26  sq.  in. 

A  E      .001  x  30,000,000 

In  general,  therefore,  where  tension  members  are  of  any  con- 
siderable length  and  distortion  under  load  is  of  importance,  they 
should  be  checked  as  above. 

10.  Compression.     If    the  member    under   consideration   be 
subjected  to  compression,  the  remarks  of  the  last  paragraph 
apply  equally  well  if  the  member  can  be  considered  a  short 
column,  i.e.,  one  whose  length  is  not  greater  than  six  times  its 
least  diameter.     If  longer  than  this  it  must  be  considered  as  a 
long  column  and  the  conditions  governing  its  design  will  be 
more  fully  treated  hereafter.     (See  Art.  20.) 

11.  Shear.  If  the  member  is  subjected  to  simple  shear  the 
expressions  for  the  relations  existing  between  the  stress,  area, 
and  load  are  similar  to  those  for  tension  or 


12.  Torsion.  If  the  member  is  subjected  to  a  torsional  stress, 
the  following  relations  exist  : 

Let  P  —load  applied  in  pounds. 
a   =  arm  of  load  in  inches. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS 


37 


Let  7p  =  polar  moment  of  inertia  of  the  section  in  biquadratic 
inches. 

ps  =  shearing  stress  in  Ibs.  per  unit  area  at  outer  fibre. 

e  =  distance  from  neutral  axis  to  outer  fibre  in  inches. 

/  =  length  of  member  in  inches. 

0  —  angle  of  deformation  in  radians. 

T  =  twisting  moment  applied  to  member  in  inch  pounds, 

Es  =  transverse  coefficient  of  elasticity. 
Then  for  torsional  strength  in  general, 

Pa  =  T  =  ^      .     .     .      .  (D) 

£ 

For  a  circular  shaft  of  solid  section, 


For  a  hollow  circular  section  whose  outside  and  inside  diameters 
are  d1  and  d2  respectively, 

p.*- 


For  deformation  under  stress  for  a  solid  circular  section,  which  is 
the  most  common  case, 


and  for  a  hollow  circular  section, 

2  Tl 


An  inspection  of  equation  (D)  shows  that  the  torsional 
resistance  for  a  given  stress  is  proportional  to  the  polar  moment 
of  inertia  divided  by  the  distance  from  the  neutral  axis  to  the 
outer  fibre.  Examination  of  equations  (E)  and  (F)  shows  that 
for  circular  sections  torsional  strength  is  proportional  to  the 
third  power  of  the  outer  diameter.  Equations  (G)  and  (H) 
show  that  torsional  deformation  is  inversely  proportional  to  the 
fourth  power  of  the  outer  diameter,  hence  torsional  stiffness  is 
directly  proportional  to  the  fourth  power  of  the  outer  diameter. 

For  a  given  amount  of  material  that  section  in  which  this  ma- 
terial is  distributed  farthest  from  the  gravity  axis  will  be  strongest 


38  MACHINE    DESIGN 

and  stiffest  as  long  as  the  walls  of  the  section  do  not  become  so 
thin  and  weak  as  to  yield  locally  from  other  causes.  The  hollow 
circular  and  hollow  rectangular  sections,  commonly  called  the 
"box  section,"  Fig.  7,  are  best  adapted,  therefore,  to  resist  tor- 


^?\ 

Z(fffffff«f«f(fffffff(f(((f(tf(jfa 


FIG.  7. 


FIG.  8. 


sional  strains.  The  box  section  is  peculiarly  useful  in  machine 
construction,  as  many  machine  members  must  carry  a  combina- 
tion of  stresses.  Machine  frames  may  be  subjected  to  tension, 
compression,  or  shearing,  combined  with  torsion,  and  the  box 
section,  while  equally  good  for  simple  stresses,  is,  as  has  been 
noted,  vastly  superior  in  torsion.  Furthermore,  the  box  section 
is  well  adapted  to  resist  combined  flexure  and  torsion.  The 
flat  sides  of  a  box  section  also  afford  facilities  for  attaching 
auxiliary  parts  and  its  appearance  is  one  of  strength  and  sta- 
bility. The  thickness  of  the  walls  being  thinner  in  hollow  than 
in  solid  forms  insures  a  better  quality  of  metal  in  castings  and 
also  more  skin  surface,  where  the  greatest  strength  of  cast  iron 
lies.  An  advantage  not  to  be  overlooked  in  some  lines  of  work 
is  the  ease  with  which  hollow  sections  can  be  strengthened  by 
increasing  the  thickness  of  the  walls  by  changing  the  core  with- 
out changing  the  external  dimensions.  The  cost  of  pattern 
work  is  about  the  same,  in  general,  for  hollow  sections  as  for  I 
or  other  sections,  while  the  work  in  the  foundry  is,  in  general,  a 
little  greater. 

Example.  A  circular  cast  iron  boring  bar  60  inches  long 
carries  a  solid  circular  boring  head  60  inches  in  diameter.  The 
bar  is  subjected  to  a  torsional  moment  of  60,000  inch  pounds 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  39 

which  is  applied  at  one  end.  It  is  desired  to  keep  the  torsional 
deflection  of  the  tool  below  -^V  when  the  bar  is  transmitting 
power  through  its  entire  length,  in  order  to  prevent  chattering 
of  the  tool.  What  should  be  the  diameter  of  the  bar  if  the 
working  stress  be  taken  as  3,000  pounds  per  square  inch  and  Es 
be  taken  as  6,000,000. 

For  torsional  strength  from  formula  E, 
60,000  X  1  6 


3,000  X- 


=  100 


For  torsional  stiffness  e  *  =  —  =  9TTr  since  e  is  in  radians 

3°  . 
and  the  length  of  an  arc  =  re,  where  r  =  radius. 

t          r    n         32  X  60,000  X  60 
.'.  from  G,  d4  =  -^—  -  —       -  i—      =  5,870 
TTX  6,000,000  x 


hence  d  =  8.8".  It  is  evident  that  the  shaft  will  be  amply  strong 
if  designed  for  stiffness,  therefore  the  last  value  would  be  used. 
If  the  section  is  made  hollow  less  metal  can  be  used.  In  this 
case  either  the  inside  or  outside  diameter  or  the  ratio  between 
them  can  be  assumed.  Let 


4  4 


whence,  df  =  and  df  —<tf  =        ^4 

Substituting  in  H, 

d*  —  d*  =  ^d*  =     32  X  60,000  X  60 
1  "  256    :        *  X  6,000,000  X  -5^15- 

.'.  dS  =  8,550  and  dl  =  9.6",  hence  d2  =  7.2". 
The  area  of  the  hollow  shaft  =  31.67  sq.  in.,  while  the  area  of 

*The  angular  deflection  or  twist  of  a  shaft  in  degrees  =5  7.  296  X   (Angular 
deflection  in  radians). 


40  MACHINE    DESIGN 

the  solid  shaft  =  60.84  s°i-  m->  so  that  with  a  small  increase  in 
diameter  one  half  the  metal  secures,  by  using  the  hollow  section, 
the  same  stiffness. 

13.  Compound    Stresses.     In   the    cases    of    simple  loading 
just  discussed  only  one  form  of  stress  is  brought  on  the  member 
and  the  design  of  the  cross-section  can  be  safely  based  on  this 
stress.     When,   however,   the  loads  applied  induce  stresses  of 
several  kinds,  it  is  no  longer  possible  in  general  to  base  the 
design  on  any  one  stress,  but  regard  must  be  had  to  the  combina- 
tion of  stresses  that  may  occur.     In  many  cases  one  or  more  of 
the  stresses  are  so  small,  or  their  action  is  such,  that  they  may  be 
neglected  in  designing  the  member,  though  they  should  always 
be  borne  in  mind.     The  stress  on  which  the  design  of  the  mem- 
ber is  based  may  be  called  the  predominating  or  primary  stress 
and  it  may  be  a  simple  stress  or  a  combination  of  simple  stresses. 
The  latter  will  be  called  a  Compound  Stress. 

14.  Flexure.     When  a  beam  is  subjected  to  simple  bending 
the  principal  stresses  that  are  induced  are  (a)  a  tension  on  one 
side  of  the  neutral  axis,  (b)  a  compression  on  the  other  side  of 
the  neutral  axis,  and  (c)  a  shearing  stress  which  acts  on  every 
section  of  the  beam  at  right  angles  to  the  tension  and  com- 
pression.    Generally  speaking,  the  shearing  stress  is  small  com- 
pared with  the  tension  or  compression  and  can  often  be  neglected. 
It  must  never  be  forgotten,  however,  and  where  the  beam  is 
designed  to  withstand  the  bending  moment  only,  care  should  be 
exercised  that  the  sections  which  are  subjected  to  a  small  bend- 
ing moment  are  not  made  so  small  as  to  yield  under  shear.     The 
predominating  stress  will  be  the  tension  or  compression  depend- 
ing on  the  material  and  the  form  of  section. 

When  a  beam  is  subjected  to  simple  flexure, — 
Let  M  =  bending  moment  at  any  section  in  inch  pounds. 

/  =  moment  of  inertia  of  section  in  biquadratic  inches. 

e  =  distance  from  neutral  axis  to  outermost  fibre  in  inches. 

A  =  deflection  at  any  point  in  inches. 

P  =load  applied  in  pounds. 

p  =  maximum  stress  at  outer  fibre  in  Ibs.  per  sq.  inch. 

E  =  coefficient  of  elasticity. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  41 

Then  for  strength,  in  general,  within  the  elastic  limit, 

pi 

if-  —  * (J) 

c 

Every  beam  when  loaded  deflects  somewhat,  depending  on  the 
shape  of  its  cross-section,  the  material,  the  way  in  which  it  is 
supported,  and  the  load  applied.  The  curve  assumed  by  a  beam 
loaded  within  the  elastic  limit  is  called  the  elastic  curve  and  is  of 
course  different  for  different  combinations  of  the  above  conditions. 
The  general  equation  of  the  elastic  curve,  whatever  the  shape  of 

d?  y       M 

the  beam  may  be,  the  load,  or  manner  of  support  is,  -7-4  =  — . 

d  oc        tL  L 

To  find  the  particular  equation  for  any  case,  M  must  be  expressed 
in  terms  of  x  and  the  expression  integrated  twice.  The  ordinate 
y,  which  is  the  deflection,  can  then  be  found  for  any  value  of  x  and 
its  greatest  value  is  the  maximum  deflection.  This  integration 
has  been  performed  for  all  the  cases  usually  met  with  in  practice, 
and  the  results  are  tabulated  in  Table  I.  It  is  to  be  noted  that 
this  tabulation  is  for  beams  of  uniform  section  and  for  stresses 
within  the  elastic  limit.  Here,  as  in  other  classes  of  machine 
members,  the  design  of  the  part  may  be  based  on  strength  or  stiff- 
ness, depending  on  the  conditions,  and  in  general  both  should  be 
considered. 

Example.  A  steel  /  beam  20  ft.  long  and  supported  at  the 
ends  is  used  as  a  track  for  a  crane  trolley  carrying  4,000  Ibs. 
Select  a  standard  rolled  /  beam  that  will  carry  the  load  with  a 
deflection  of  not  more  than  yV  at  the  centre  and  a  maximum 
stress  of  not  more  than  8,000  Ibs. 

From  Table  I, 

3"        PI3  4,000  X  2403 


A    = 


16      48  E  I      48  X  30,000,000  X  / 


4,000  X  2403  X  1 6 

whence  /  =  — =  205. 

48  X  30,000,000  X  3 


*  The  expression  —  is  sometimes  called  the  modulus  of  the  section  and  is 
generally  indicated  by  the  letter  Z.  It  should  he  noted,  however,  that  this  ex- 
pression is  applicable  only  to  symmetrical  sections  as  e  may  have  two  values  for  other 

—  is  termed  the  resisting  moment. 


42  MACHINE    DESIGN 

From  handbooks  on  structural  shapes  it  is  found  that  the 
moment  of  inertia  of  a  12"  /  beam  weighing  31.5  Ibs.  per  foot  is 
215.8.  Let  such  a  beam  be  chosen.  Then  from  formula  J,  the 

Me       2,000  X  io  X  12  X  6 
stress  p  =  —:—  =  -          -  =  7,000  Ibs.  nearly.    The 

0 

section  therefore  is  satisfactory. 

15.  Beams  of  Uniform  Strength.  The  values  in  Table  I 
refer  to  beams  of  uniform  cross-section.  In  nearly  all  cases  the 
bending  moment,  which  is  usually  the  basis  of  design,  varies  and 
if,  therefore,  the  beam  is  made. strong  enough  at  its  most  strained 
section  and  uniform  in  cross-section  throughout  its  length  it  will 
have  an  excess  of  material  at  every  other  section.*  Sometimes  it 
is  desirable  to  have  the  cross-section  uniform,  while  in  other 
cases  the  metal  can  be  so  distributed  that  every  section  shall 
have  the  necessary  strength  to  resist  the  bending  moment  and  no 
more.  In  the  latter  cases  the  shearing  stress  must  be  looked  after 
carefully.  Table  II  gives  a  few  of  the  forms  most  usually  met 
and  an  example  may  make  their  application  clear. 

Example.  A  cantilever  of  rectangular  section  30  inches  long 
carries  at  its  outer  end  a  load  of  1,000  Ibs.  It  is  to  have  a  uni- 
form thickness.  What  is  its  vertical  outline  so  as  to  have  uni- 
form strength  ? 

Let  the  thickness  =  £>  and  the  variable  height  =  y.  Then  the 
moment  at  any  section  at  a  distance  x  (Fig.  I,  Table  2)  is  Px, 
and  this  must  be  equal  to  the  resisting  moment  of  the  section  at 
each  point,  hence 

pi      pby*          2      6Poc 

P  x  =  —  =  -^-  or  r  =  — — 
e  6  pb 

which  is  the  equation  of  a  parabola  whose  vertex  is  at  the  outer 
end  of  the  beam.  In  the  problem  assumed  let  b  =  1.5  inches  and 
let  p  =  4,000  Ibs.  Then  when  #  =  30,  y  —  h  =  5.5*.  In  a  similar 
way  other  points  may  be  found  or  the  curve  may  be  laid  out  by 
graphical  method.  The  shearing  load  at  any  point  is  P,  and 
hence  the  shearing  stress  increases  as  the  cross-section  of  the 

*  This  of  course  does  not  cover  the  possible  case  where  the  effect  of  shearing 
or  other  stresses  may  exceed  that  due  to  flexure. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  43 

beam  decreases.  When  oc  =  o,  y  =  o,  and  in  general  when  x 
is  small,  y  is  very  small;  therefore  the  outer  end  of  the  member 
must  be  modified  so  as  to  safely  carry  the  shearing  stress.  Refer- 
ence will  be  made  to  this  again  under  the  section  dealing  with 
machine  attachments  (see  chap.  16).  It  is  to  be  especially  noted 
that  these  theoretical  shapes  are  based  on  certain  assumptions 
and  unless  these  are  observed  in  the  design,  the  theoretical  out- 
lines do  not  apply.  Thus  in  the  cantilever  example  above,  if  the 
thickness  of  the  beam  is  not  kept  uniform  the  outline  for  uniform 
strength  is  not  a  parabola.  The  mistake  of  using  a  parabola 
when  the  thickness  is  not  uniform  is  often  made  when  /  or  T 
sections  are  used  instead  of  uniform  thickness  or  depth.  It  is 
evident,  that,  whatever  may  be  the  form  of  section  adopted,  by 
means  of  the  bending  moment  and  shearing  load  the  correct 
depth  of  section  can  be  found  for  a  number  of  points  and  a 
curve  plotted  that  will  answer  the  requirements  of  uniform 
strength. 

1 6.  Combined  Flexure  and  Torsion.  Let  the  force  ?,  Fig.  8, 
act  upon  a  rod  with  an  arm  a  at  a  distance  from  the  support  equal 
to  /.  Then  the  stresses  induced  in  the  section  close  to  the  support 
are 

(a)  flexure  due  to  the  bending  moment  PI 

(b),  torsion    "  "    "     twisting       "        Pa 

p 

(c)  shearing  "         "     direct  load  and  equal  to  —.- 

A. 

The  shearing  stress  is  usually  very  small  compared  to  that  due 
to  bending  and  twisting,  and  can  be  neglected;  the  predominat- 
ing stress  therefore  is  that  due  to  the  combined  action  of  the 
bending  and  twisting  moments. 

It  can  be  shown  that  if  a  bar  or  rod  is  subjected  to  a  longi- 
tudinal tensile  or  compressive  stress  and  at  the  same  time  to  a 
shearing  stress  at  right  angles  to  its  length,  the  combination  of 
these  stresses  may  produce  similar  stresses  greater  than  either 
and  acting  along  planes  other  than  those  along  which  the  original 
stresses  act.* 

*  Church's  "Mechanics,"  page  317. 


44 


MACHINE    DESIGN 


TABLE  I 

BEAMS   OF  UNIFORM  SECTION 


Diagram  of  Loads, 

Bending  Moments 

and  Shear 


Greatest 

Bending 

Moment 

M 


Location 
of 
M 


Greatest  [Location 
Deflection       of 

A 


Maximum 

Shearing 

Force 


Section 
where 
Shear 
s  Max 


pi3 

3EI 


From 
CtoB 


Wl3 
8EI 


wl  =  W 


)OOOOOOOA 


Wl2 


•fpl 


wi  +  P 


II 


_  5 


16 


PI 


PI- 


PI3     At 4- 0-451 
107  El  from 

A 


JL  p 

16 


BtoC 
CtoA 


9wl2 
~~~ 


wi* 
185EI 


tf=load  per  unit  length.    W  =  total  distributed  load.    P  =  concentrated  load. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS 
TABLE  I— Continued 


45 


?t'=load  per  unit  length.     W=  total  distributed  load.    P=  concentrated  load. 


46 


MACHINE    DESIGN 


TABLE  I— Continued 


Diagram  of  Loads 
Bending  Moments 
and  Shear 


Greatest 

Bending 

Moment 

M 


Loca- 

tion of 

M 


Greatest 
Deflection 


Loca- 
tion of 


Maximum 

Shearing 

Stress 


ection 
where 
Shear 
s  Max. 


B  C  A 

noonnooo 


III 


Supported 
both  En 


48"E  I 


W 


AorB 


et 
a 


81 V' 

LT 


Pa 


DtoC 


in  a 


B   D  E  C  A 

orbnnnnoorxho 


K 


CandD 


andD 


Unif 
Sym 
S 


PI 


XVI 


ed  at 
One 
at  C 


at 

A.B. 
andC. 


PI3 
192  El 


BtoC 
CtoA 


ed  at  both  en 
niform  Load 


Aor.B 


Wl* 

384  El 


AorB 


ed  a 
nifo 


at-B 

atC 

at  A 


3EI(l+SUi 


«>=load  per  unit  length.     W= total  distributed  load.   P=  concentrated  load. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  47 


TABLE  II 

BEAMS  OP  UNIFORM  STRENGTH 


Outline 

of 
Beam 


Greatest 

Bending 

Moment 

M 


Loca 
tion  o 
M 


Greatest 
Deflection 


tion  of 

A 


Maximum  |8ection 
Shearing 
Stress 


& 


PI 


8P13 
Ebh3 


Elevation 


_..__amof 

Moments  and  Shear 

same  as  I 


ilJ 

|*« 


PI 


6P13 
Ebh3 


Diagrams  as 
in  No.  Ill  Table  I 


W 


••3    N     N 


V     I 


1- — H 

Diagrams  same 
as  No.  X  Table  I 


when  I]»l2 

A  - 
_Pli_ 


vi  r  -*t 


Diagrams  same 
as  No.  XII  Table  I 


Centre 


vii  : 


Diagrams  same 
as  No.  IX  Table  I 


PI 


Centre 


w=load  per  unit  length.     TT=  total  distributed  load.   P=  concentrated  load. 


48  MACHINE    DESIGN 

If  /  be  the  greatest  direct  tensile  or  compressive  stress  and  s 
the  greatest  direct  shearing  stress  applied  to  the  bar,  then  the 
maximum  tensile  or  compressive  stress  p  due  to  /  and  5  is  given 
by  the  following  equation  : 

/>  =  JP  +  V/MTJ?]        •      •      .      •      (i) 
and  the  maximum  shearing  stress  pB  due  to  /  and  s  is 


It  is  evident  that  the  numerical  value  of  p  will  always  exceed 
that  of  pB  and  therefore  if  the  material  used  has  approximately 
the  same  tensile  and  shearing  strength  the  design  can  be  safely 
based  on  (i).  But  should  the  allowable  shearing  strength  of  the 
material  be  less  than  the  tensile  strength,  as  is  usually  the  case, 
it  may  happen  that  the  shearing  stress  pa  as  found  by  (2)  would 
dictate  a  larger  section  than  that  required  by  p  as  found  by  (i). 

If  the  tensile  stress  is  due  to  a  bending  moment  and  the  shear- 
ing stress  is  due  to  a  twisting  moment  the  values  of  s  and  /  can 
be  found  from  equations  J  and  D  respectively  and  p  and  pt 
obtained  as  above  in  equations  (i)  and  (2)  respectively. 

Example.  A  certain  section  of  a  circular  cast  iron  shaft  is 
subjected  to  a  bending  moment  M  of  10,000  inch  Ibs.  and  a 
twisting  moment  T  of  60,000  inch  Ibs.  The  allowable  tensile 
stress  p}  is  2,000  Ibs.  per  square  inch  and  the  allowable  shearing 
stress  ptJ  is  1,600  Ibs.  per  square  inch.  It  is  required  to  design 
the  cross-section  of  the  shaft. 

Me       32  M      32  X  10,000      100,000 
From  /,  /  .     —        —p  -         gxrf,  —p-    nearly 

Te      i6T       16  X  60,000       300,000 

and  from  D,   s  =  —  =  —=  =  -  —  5  -  =  —  —^  —  nearly 
L          x  d  ~  d  d 


hence  from(i),/>,  =     ~  —  and  since  p  =  2,000 


2,000 

or  d  =  5-  55" 


From(2),/>8  =  —~^  and  since  pt  «*  1,600,  d*  =  3^°^°. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  49 

.'.^  =  5. 8"  or  X"  greater  than  that  given  by  (i).  It  is  evident 
that  the  last  value  should  be  taken. 

Equations  (i)  and  (2)  are  general,  and  applicable  to  any  and  all 
sections,  but  for  circular  shafts  operating  under  conditions  that 
produce  both  bending  and  twisting  it  has  been  found  convenient 
to  make  use  of  what  may  be  called  an  equivalent  or  ideal  bending 
moment  which  may  be  derived  from  equation  (i)  as  follows. 

Let  Me  =  the  equivalent  bending  which  will  produce  the 
maximum  direct  stress  p. 

Let  M  =  the  bending  moment  producing  the  direct  stress  t. 
"  T  =   the  twisting  moment  producing  the  shearing  stress  s. 
"  r     =  radius  of  shaft. 

From  J,M=  ~  and  Me  =  -^, 

and  from  Z>,  T  =  ~p  =  — 
r  r 

(Since  /p  =  2/  for  circular  <or  other  sections  for  which  the  mo- 
ments of  inertia  about  two  perpendicular  axes  are  equal.) 

Multiply  equation  (i)  through  by  — ,   whence 


M    r 


e  =   y2M  +  K  V  M2  +  T2  .      .     .  (K) 

In  a  similar  manner   an   equivalent  twisting  moment  can  be 
deduced  from  (2)  thus, 


The  quantities  M  and  T  are  usually  large  and  the  numerical 
work  involved  in  solving  K  and  Kl  can  be  simplified  by  writing 
M  =  x  J1,  where  x  for  any  particular  problem  will  be  a  known 
quantity. 

Whence  K  reduces  to, 

Me  =  K  T[x  +  Vx*+  i]      .... 
and  X"t  reduces  to, 


50  MACHINE    DESIGN 

It  is  to  be  especially  noted  that  Me  and  Te  are  equivalent  mo- 
ments in  a  numerical  sense  only;  that  is,  if  a  bending  moment 
M  and  a  twisting  moment  T  are  applied  to  a  shaft,  producing  a 
tensile  stress  /  and  a  shearing  stress  s  respectively,  then  Me  is 
a  bending  moment  which  will  give  a  stress  equal  to  the  maximum 
resultant  tensile  or  compressive  stress  />,  and  Te  is  a  twisting 
moment  which  will  give  a  stress  equal  to  the  maximum  resultant 
shearing  stress  pa,  reference  being  made  to  the  same  section. 

The  application  of  these  equations  to  the  investigation  of  any 
existing  shaft  subjected  to  a  bending  moment  M  and  a  twisting 
moment  T  is  obvious,  and  it  remains  to  consider  their  applica- 
tion to  the  design  of  new  shafts.  It  has  been  pointed  out  that 
the  greater  numerical  value  given  by  equation  (i)  does  not  nec- 
essarily indicate  that  a  larger  section  will  result  from  its  adoption 
than  would  result  from  the  use  of  equation  (2).  For  the  same 
reasons  the  greater  numerical  value  of  Me,  obtained  from  K  may 
not  give  a  larger  section  than  would  be  obtained  from  Te  by  ap- 
plying Kr  It  is  necessary  therefore  to  determine  under  what 
conditions  each  should  be  used  for  designing  in  order  that  the 
maximum  diameter  of  shaft  shall  be  found  in  all  cases. 

pi      pnd* 
From  /,  M    -  — 


r  32 

M  '        16       2  M  P 
.  . 


In  a  similar  way  from  E 


l6  v   T< 

'    ~ 


Since  in  any  given  problem  M  and  T  are  always  known,  Me 
and  Te  can  always  be  found  from  K  and  Kl  (or  K2  and  K3)  and 
since  the  allowable  values  of  p  and  pa  can  always  be  assigned,  the 
diameter  of  the  shaft  d  can  always  be  determined  from  both 
equations  (3)  and  (4)  and  the  larger  value  selected  as  in  the  problem 
previously  solved.  It  is,  however,  desirable  to  know,  for  any 
set  of  conditions,  whether  equation  (3)  or  equation  (4)  will  give  the 
greater  value  of  d  without  the  necessity  of  solving  both  equa- 
tions. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  51 

It  is  evident  that  in  order  that  equations  (3)  and  (4)  may  give  the 

same  diameter  of  shaft  -  — - -  must  equal  — i  or  — —  =  —  and 

P  P.       2M.       p 

that  for  conditions  other  than  these,  either  equation  (3)  or  equa- 
tion (4)  may  give  the  greater  diameter.  It  is  therefore  neces- 

T1  -/> 

sary  to   investigate  the  relations  existing  between  — —  and  — 

for  three  sets  of  conditions. 

(1)  When  equations  (3)  and  (4)  will  give  equal  values  of  d. 

(2)  "       equation   (3)  will  give  the  greatest  value  of  d. 

f^\  «  «  (V\       «          «       «  u  u          u    ti 

(i)  It  has  already  been  shown  that  equations  (3)  and  (4)  will  give 
equal  values  of  d  when  — ?  =       *    or  if  — -8  be  called  y9  then 

Te  VM2+T 

~  2Me~  M  +  V  M2+T* 
is  the  equation  of  a  curve  which  expresses  all  the  simultaneous 

.  rrt 

values  of  —  and  — ~  for  which  equations  (3)  and  (4)  will  give 

equal  values  of  d.  The  value  of  either  M  or  T  in  equation  5  may 
vary  from  zero  to  infinity  and  the  most  convenient  way  of  plotting 
simultaneous  values  of  M  and  T  is  to  plot  their  ratio.  If  then, 
in  equation  (5),  the  relations  as  given  in  K2  and  K2  be  substi- 
tuted for  those  in  K  and  Kv  the  equation  becomes 


Te  W+  i  . 

...      (6) 


which  is  the  equation  of  a  curve  expressing  all  the  simultaneous 
values  of  y  (or  —  J  and  x  (or  —  j  for  which  equations  K2  and 

^T3  will  give  equal  diameters  of  shaft. 

It  is  desirable  before  plotting  the  curve  to  examine  the  limits 
between  which  x  and  y  may  vary.  It  is  clear  that  for  M  =  o 
x  =  o,  and  for  T  =  o  x=  oo  ,  hence  the  limits  of  x  are  o,  and  oo  . 


MACHINE    DESIGN 


Using  these  same  limits  for  M  and  T  in  equation  (5)  it  is  found 
that 


when  M  =  o,  y 


i  and  M  =  — 

2 


and  when  T  =  o,  y  =  J  and  Me  =  Te 

That  is  for  all  materials  where  the  ratio  of  allowable  shearing 
to  tensile  stress  lies  between  i  and  y2  there  are  always  simulta- 
neous values  of  M  and  T  for  which  equations  (3)  and  (4)  will  give 
equal  values  of  d.  The  curve  giving  these  simultaneous  values 
is  shown  in  Fig.  9  and  has  been  plotted  from  equation  (6). 

(2)  If  for  any  given  value  of  —  within  the  limits  i  and  X  a 


.9 

.,    '8 

;io, 

t.7 
.6 
A 

\ 

FieU  of  K, 

\ 

\ 

M 

8  = 

.w 

i 

X2 

4-  I 

r 

Fi 

eld 

of 

£t 

x 

lA 

"i 

^ 

-- 

"~-~ 

^-^^_ 

\ 

•^"-^- 

•  . 

—  —  — 

—   •. 

—  — 

.4    .5    .6    .7     .8    .9     1    1.1  1.2  1.3  1.4  1.5  1.6  1.7  1.8  1.9    2    2.1  2.2 

"I 

FIG.  9. 


ratio  of  —  be  taken  greater  than  the  simultaneous  value  given 

by  the  curve  (or  in  other  words  if  the  co-ordinates  chosen  inter- 
sect above  the  curve)  equation  (3)  will  give  the  largest  value  of  d. 

M 
For  the  value  of  —  can  be  increased  only  by  making  M  greater 

relatively  to  T  and  an  examination  of  K  and  Kl  shows  that  in- 
creasing M  increases  K  more  rapidly  than  it  does  K^  Hence 
in  such  cases  K  (or  K2)  applies  and  equation  (3)  which  is  based 
on  them  will  give  the  largest  value. 

Further,  for  values  of  —  equal  to  or  greater  than  unity,  equa- 

tion (3)  will  also  give  the  largest  value  of  d.     For  it  has  just  been 

T 

shown  that  Me  can  never  be  less  than  —  •  and  only  equals  this 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  53 

when  M  =  o.    For  all  finite  values  of  M,  therefore,  Mt  must  be 
greater  than  —\    and  it  is  evident  from  equations  (3)  and  (4)  that 

T 

for  values  of  pt  =  p  and  Me>  —  equation  (3),  which  is  based  on 

K  (or  KI),  will  give  the  greatest  value  of  d. 

(3)  In  a  similar  way  it  can  be  shown  that  for  all  simultaneous 

values  of  -7  and  —  which  intersects  below  the  curve  and  within 
P  T 

the  limits  y  =  i  and  y=lA\  or  for  all  materials  where  —  is   less 

P 
than  X>  equation  (4),  which  is  based  on  Kl  (or  Ks),  will  give  the 

greatest  value  of  d. 

Summary.  Equations  K2  and  K3  are  the  most  convenient 
forms  of  equivalent  moments  and  will  be  used  in  this  work.  It 
is  to  be  particularly  noted  that  they  are  applicable  only  to  circu- 
lar or  other  sections  where  the  polar  moment  of  inertia  is  equal 
to  the  sum  of  the  rectangular  moments  of  inertia  around  perpen- 
dicular axes  (see  page  50).  Where  the  section  to  be  considered 
is  more  complex  the  solution  must  be  based  on  the  original  equa- 
tions (i)  and  (2)  in  a  similar  manner  to  that  employed  in  the  ex- 
ample on  page  49.  Equation  K2  should  be  used  where  the 

simultaneous  values  of  —  and  —  intersect  above  the  curve  which 
P  T 

is  always  the  case  whenever  — ->   i.     Equation  K3  should  be 

used  where  the  simultaneous  values  of  — -  and  —  intersect  below 

P  T 

the  curve,  which  is  always  the  case  whenever  — -  <  J. 

Example  i.  An  engine  cylinder  is  i6"X24//  (piston  16"  in 
diameter  and  stroke  of  24"),  steam  pressure  =  100  Ibs.  per  square 
inch.  The  centre  of  the  crank  pin  overhangs  the  centre  of  the 
main  journal  by  15"  (measured  parallel  to  the  axis  of  shaft). 
Assume  that  the  pressure  on  the  crank  pin  may  be  equal  to  100 
Ibs.  unbalanced  pressure  per  square  inch  of  the  piston  when 
the  connecting-rod  is  perpendicular  to  the  crank  radius.  Allow- 


54  MACHINE    DESIGN 

ing  8,000  pounds  as  the  maximum  allowable  direct  stress  and 
6,400  as  the  maximum  allowable  shearing  stress,  compute  the 
diameter  of  the  shaft. 

Area  of  piston  =  200  sq.  inches;  radius  of  crank  (arm  of 
maximum  twisting  moment)  r  =  12";  arm  of  bending  moment 
a  =  15" 

T  =  200  X  ioo  X  12  =  240,000  inch  Ibs.     Also 

M  =  200  X  ioo  X  15  =  300,000 

M  p,       6,400 

*  =  ^  -  15  +  »  -  1.25;    j  -  g^  -  -8  =  y. 

By  referring  to  Fig.  9  it  is  seen  that  for  y  =  — -  =  .8  and  x  = 

1.25  the  ordinates  intersect  above  the  curve,  hence  K2  should  be 
used. 


From  K2y        Me  =  J  [1.25  +  V  1.25*  +  i]  240,000 
=  342,000  inch  Ibs. 


Example  2.  —  A  circular  cast  iron  shaft  is  subjected  to  a  twist- 
ing moment  of  250,000  inch  Ibs.  and  a  bending  moment  of  62,500 
inch  Ibs.  The  allowable  tensile  stress  is  2,000  Ibs.  per  sq.  inch 
and  the  allowable  shearing  stress  1,400  Ibs.  Determine  the 
diameter  of  the  shaft. 


TT  P*        I^4oo  62,500 

Here  ;y  =  —  =  —    -  =  .7  and  x  =  -         -  =  .25. 
p         2,000  250,000 

From  the  curve,  Fig.  9,  it  is  seen  that  for  y  =  —  -  =  .7  and 

x  =  .25  the  intersection  of  the  ordinates  falls  below  the  curve, 
hence  K~  should  be  used. 


Then  Te  =  [</<.2$)a  +  i]  250,000  =  257,500 
*  =  l6X257-5°°  -  935 

7T    X    1,400 

d  =  9.75.  inches. 
Suppose,  however,  that  K2  should  be  used. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS 
Then,       Me  =  \  [.25  +  V(.2$)2  +  i]  250,000  =  160,000 

=  830 


55 


7T  X   2,000 

.'.  d  =  9.4  inches  or  .35"  less  than  the  value  given  by  K2. 

A  convenient  graphical  solution  of  K2  and  K3  is  shown  in  Fig. 
9  (a)  which  may  be  used  as  follows: 

For  K3  make  Oa  —  unity;  lay  off  Ob  =  x  to  the  same  scale  on 
the  vertical  axis.  Draw  ab  extending  it  beyond  b  for  a 
length  somewhat  greater  than  x; 

then  ab  =  V  Ob2  +  Oa2  =  VV  +  i ; 
hence  Te  =  a  b  X  T. 

For  equation  K2,  lay  off  be  =  Ob  =  x  along  the  extension  of 
ab.     Then bc  +  ab  =  ac  =  ~ 

Hence  Mt  =  K  [ac  X  T]. 


Unity . . >j 


17.  Other    Formulas.  —  Equation    K    is    sometimes    trans- 
formed into  an  equivalent  twisting  moment.     Since  in  general 


M 


—  and  71  = 


—  ,  for  an  equal  intensity  of  stress  (that 

is,  pB  =  p)  T  =  2M  for  the  same  section.  If  therefore  it  is  con- 
sidered more  convenient  to  use  an  equivalent  twisting  moment 
instead  of  an  equivalent  bending  moment  it  is  allowable  to  sub- 
stitute for  Me  (the  bending  moment,  equivalent  to  the  combined 
bending  and  twisting  moment),  l/£  Te  (a  twisting  moment  equiva- 
lent to  the  combined  bending  and  twisting  moments)  provided 
the  same  allowable  direct  stress  is  used  with  Te  in  solving  for  the 
diameter  of  shaft. 

.  *.  r  =  2  M  =  M 


56  MACHINE    DESIGN 

Equations  K^  K3,  and  K4  are  all  different  forms  of  Rankine's 
formula  for  combined  bending  and  twisting.  Other  authorities 
give  slightly  different  coefficients.  Thus  Grashof  gives 

'M9  =  |  M  +  I  v'M2  +  T  ....  (7) 
While  others  give 

Me  =  0.35  M  +  o  .65  VM2  +  T2  .  .  (8) 
The  diameter  of  shaft  given  by  equations  (7)  and  (8)  will 
not  differ  much  from  that  given  by  K2,  for  any  set  of  conditions, 
except  where  the  bending  moment  is  very  small.  At  the  limit 
where  the  bending  moment  M  is  equal  to  zero,  Grashof  s  formula 
gives  a  value  of  Me,  25  per  cent  greater  than  that  given  by  K2.  But 
it  may  be  noted  that  in  general  for  all  materials  whose  shearing 
strength  is  less  than  their  tensile  strength  (and  this  is  the  case 
for  most  materials  used  in  engineering)  that  when  M  is  small  or, 
in  other  words,  when  the  shearing  stress  predominates,  it  is 
safer  to  use  K3  in  preference  to  K2.  It  will  be  found  that  for 
the  range  where  equations  (7)  and  (8)  give  values  greater  than 
K2,  that  these  values  will  still  be  less  than  those  obtained  from 
K3  or  at  least  not  enough  greater  to  warrant  the  use  of  a  different 

formula  in  place  of  Ky     Take  for  example  steel  where  —  =  .8 

and  x  =  .1,  which  is  down  close  to  the  limit  where  Grashof 's 
formula  gives  the  greatest  value  compared  to  K2.  Expressing  d3 
in  terms  of  T  as  in  equations  (3)  and  (4) , 

3       ,-1= 

from  K2  d  =  1.77  *j  — 

J£ 

from  K3  d  =  1.84  J  — 
\  p 

3      (Y 

from  Grashof's  formula     d  =  1.88  -J — , 

\  p 

from  which  it  is  seen  that  the  difference  between  d  as  determined 
by  K3  and  Grashof's  formula  is  negligible.  The  same  evidently 
applies  to  equation  (8)  which  differs  but  little  from  Grashof's. 
As  the  value  of  x  increases,  the  difference  between  these  equiva- 
lent bending  moments  decreases,  and  any  variation  is  more  than 
covered  by  the  factor  of  safety  which  must  be  used. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  57 

1 8.  Combined  Torsion  and  Compression.  Propeller  shafts 
of  steamers  and  vertical  shafts  carrying  considerable  weight  are 
subjected  to  combined  twist  and  thrust.  The  span,  or  distance 
between  bearings,  is  frequently  so  small  that  the  shaft  may  be 
considered  as  subjected  to  simple  compression,  so  far  as  the 
action  of  the  thrust  is  concerned. 

The  intensity  of  this  compressive  stress  in  such  cases  is 

c-£ 

~  *<?' 

in  which  P  =  the  thrust,  and  d  =  the  diameter  of  the  (solid  cir- 
cular) shaft. 

If  T  =  the  twisting  moment  on  the  shaft,  r  =  %  d  =  the 
radius  of  the  shaft,  /p  =  the  polar  moment  of  inertia  =  2  / 
(=  2  times  the  rectangular  moment  of  inertia),  and  s  =  the 
intensity  of  shearing  stress  due  to  T,  then 

T  -  —*  -  4Sl  Td         l6r 

r  d       '"•  S    ;  7/~   =  *d3 

for  solid  circular  shafts. 

The  resultant  maximum  stresses  are  those  due  to  the  com- 
bined actions  of  a  normal  stress  (compression)  and  a  tangential 
stress  (shear)  as  in  the  case  of  combined  bending  and  twisting 
(Art.  17);  hence  equations  (i)  and  (2)  of  the  preceding  article 
apply  and  may  be  used  to  find  the  maximum  compressive  or  max- 
imum shearing  stress;  or  if  c  be  the  compressive  stress  due  to  P,  s 
be  the  shearing  stress  due  to  T,  pc  the  maximum  resultant  com- 
pressive stress,  and  pn  the  maximum  resultant  shearing  stress,  then 


or  p  = 


58  MACHINE    DESIGN 

It  is  difficult  to  find  the  value  of  d  for  a  given  value  of  pc  or  ps 
from  the  above  equations,  and  it  is  much  more  convenient  to 
assume  a  trial  diameter  d  and  then  check  for  the  values  of  pc 
and  p&  to  see  that  they  do  not  exceed  the  allowable  compressive 
and  shearing  stresses  of  the  material  under  consideration. 

If  the  span  of  the  shaft  between  bearings  is  so  great  that  the 
shaft  must  be  considered  as  a  column  likely  to  buckle,  the  trial 
diameter  of  the  shaft  may  be  taken  so  as  to  bring  the  mean 
compressive  stress  c  well  below  the  allowable  value,  and  after 
solving  for  pc  and  pa  the  shaft  may  also  be  checked  as  a  long 
column  (Art.  20).  In  steel  shafting  it  is  necessary  usually  to 
apply  equation  (L)  only,  but  it  is  well  to  check  the  shearing 
stress  pn  against  the  allowable  stress  by  applying  (L,). 

19.  Flexure  Combined  with  Direct  Stress.  If  the  section 
XY,  Fig.  10,  be  acted  on  by  a  force  P  at  a  distance  from  its 
gravity  axis  O  equal  to  a,  the  stresses  induced  in  the  section  will 
be:— 

(a)  A  uniformly  distributed  stress  due  to  the  load  P  and 

p 

equal  to  —  per  unit  area.     This  will  be  tensile  or  compressive, 

A. 

depending  on  the  direction  of  P. 

(b)  A  flexural  stress  due  to  the  bending  moment  P  a.     This 
flexural  stress  will  be  a  tensile  stress  on  one  side  of  the  gravity 
axis  which  is  at  right  angles  to  a,  and  compressive  on  the  other. 

If  the  direct  stress  induced  in  the  section  by  the  load  P  is 
tensile,  then  the  flexural  stress  on  the  side  toward  the  load  is 
tensile.  If  the  direct  stress  induced  is  compressive,  the  flexural 
stress  on  the  side  toward  the  load  is  compressive.  The  maxi- 
mum stress  will  be  the  greatest  algebraic  sum  of  these  combined 
stresses  at  the  outer  fibres  at  X  or  Y.  The  distribution  of  these 
stresses  for  both  cases  is  shown  graphically  in  Fig.  10,  where 
tensile  stresses  are  plotted  above  the  line  U  V  and  the  compress- 
ive stress  below;  the  ordinates  under  r  s  representing  the  flexural 
stresses,  and  those  under  m  n  the  direct  stresses.  An  inspection 
will  show  where  the  algebraic  sum  is  greatest.  In  the  case  shown 
the  combined  compressive  or  combined  tensile  stresses  at  X  are 
the  greatest  which  may  come  on  the  section,  depending  on  the 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS 


59 


direction  of  P.     This  is  not  necessarily  so,  as  a  brief  reflection 
will  show  that  if  O  be  located  near  enough  to  X  the  reverse  of 
the  above  conditions  may  exist.     The  form  of  section  and  location 
of  the  gravity  axis  should  be  fixed  with  reference  to  the  relative 
tensile  and  compressive  strength  of  the  material  used. 
Let  p'  =  the  direct  stress  due  to  P, 
Let  p"  =  the  tensile  or  compressive  stress  due  to  P  <z, 
Let  p    —  the  maximum  stress  in  the  section  at  X  or  Y. 

Then  from  formula  A ,  p'  =  — ,  and  from  formula  /,  p"  = 

A  I 


Therefore, 


Pae* 


(M) 


FIG.  10  (a). 


FIG.  10  (b). 


where  e  is  the  distance  from  O  to  the  outer  fibre  at  either  X  or 
Yj  depending  on  which  is  under  consideration.  If  the  material 
used  is  equally  strong  in  tension  and  compression  the  gravity 
axis  should  not  be  far  from  central,  but  where  cast  iron  is  used 
it  is  advantageous  to  distribute  the  metal  more  toward  the 
tension  side,  thus  drawing  the  gravity  axis  toward  that  side. 
This  increases  e  on  the  compression  side,  and  hence  increases  the 

*See  Church's  "Mechanics,"  p.  362. 


60  MACHINE    DESIGN 

compressive  stress.  It  decreases  e  on  the  tension  side,  and  hence 
decreases  the  tensile  stress.  Cast  iron  is  much  stronger  in 
compression  than  in  tension,  and  therefore  a  greater  moment  can 
be  withstood  by  a  given  cross-sectional  area  when  distributed  in 
this  manner. 

It  is  not  practicable,  in  general,  to  solve  equation  (M)  for  th& 
direct  determination  of  the  dimensions  of  a  cross  section  to  sus- 
tain a  given  eccentric  load  P  with  an  assigned  intensity  of  stress 
/>,  because  both  A,  /,  and  e  are  functions  of  the  required  dimen- 
sions; and  with  any  but  the  simplest  sections  complicated  func- 
tions result.  With  solid,  square,  or  circular  sections,  or  in 
general  where  only  one  dimension  is  unknown,  it  is  possible  to 
reduce  M  to  a  form  which  can  be  solved;  but  the  algebraic  ex- 
pression is  a  troublesome  cubic  equation.  The  practical  way  is 
to  assume  a  trial  section  and  check  this  for  P  or  p. 

Example  i.  A  small  crane  (Fig.  n)  has  a  clear  swing  of  28 
inches.  The  section  at  m  n  is  shown  by  Fig.  1 1  (b) .  Find  the 
load  corresponding  to  a  maximum  fibre  stress  (compression)  of 
9,000  Ibs.  per  square  inch  at  n. 

*    L  +  P^  p      PAI 

P  "  A  "       /  "  I  +  Aae 

a  =  28  +  2  =  30        A  =  2  X  4  —  1.5  X  3-  =  3-5 

/  =  TV(2  x  64-  1.5  +  27)  =  7.3 

.-.  p  =  9,000x3.5x7.3  =I?06Qlbs. 

7-3  +  3-5  X3°X  2 

Example  2.  A  punching  machine  (Fig.  12)  has  a  reach  of 
22  inches.  Maximum  force  P  acting  at  the  punch  is  taken  at 
70,000  Ibs.  Design  the  section  m  n  so  that  the  maximum  fibre 
stress  at  n  (tension)  shall  be  about  2,400  pounds  per  square  inch, 
and  check  the  compressive  stress  at  m. 

The  general  form  of  section  best  adapted  to  this  case  is  that 
shown  in  Fig.  12  (b).  Taking  the  trial  dimensions  as  in  Fig.  12  (6), 
the  neutral  axis  is  found  to  be  8"  from  n. 

.*.  a  =  22  +  8  =  30.  It  is  also  found  that  A  =  216  and 
/  =  7,680. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS 


6l 


P    = 


at  n, 
70,000 


70,000  X  30  X  8 


216 


=  325  +  2,200  =  2,525    Ibs. 


and  at  m, 


7o,ooo 
216 


70,000  X  30  X  ii 
7,68o 


The  tensile  stress  is  slightly  greater  than  the  limit  assigned. 
If  this  excess  is  not  considered  permissible  a  stronger  section 
must  be  taken.  It  is  evident  that  this  can  be  accomplished  by 
massing  the  metal  still  further  toward  the  tension  side  as  the 
compressive  stress  is  very  low. 


FIG.  ii. 


FIG.  12. 


The  theory  regarding  the  position  of  the  neutral  axis  given 
above  is  that  in  general  use  for  such  cases.  Recent  investiga- 
tions have  pointed  out  the  fact  that  this  theory  is  not  absolutely 
rigid  for  curved  beams.  For  the  usual  case  of  design  it  is  be- 
lieved that  the  above  is  sufficiently  accurate. 

20.  Stresses  in  Columns  or  Long  Struts.  When  a  short 
bar  is  subjected  to  an  axial  compressive  load  the  stress  induced 
in  each  section  is  simple  compression  (see  Art.  20),  and  the  value 
of  the  stress  p  is  given  by  formula  (A)  or 

W 
P-T. 

If,  however,  the  bar  is  more  than  4  to  6  times  as  long  as  its  least 
diameter,  the  above  equation  does  not  apply,  as  the  bar  will,  if 


62  MACHINE    DESIGN 

proportioned  as  above,  deflect  laterally  under  the  load  and  will 
ultimately  break  under  a  compound  stress  due  to  compression 
and  lateral  bending.  Such  a  member  is  called  a  column. 

Theoretical  equations  for  the  design  of  columns  were  first 
developed  by  Euler.  Other  formulae  were  later  developed  ex- 
perimentally by  Hodgkinson  and  Tredgold.  Gordon  and  Rankine 
have  also  proposed  equations  for  the  design  of  this  class  of  mem- 
bers. The  student  is  referred  to  any  good  treatise  on  the  Me- 
chanics of  Materials  for  a  fuller  discussion  of  these  expressions 
than  can  be  given  in  this  work. 

^  •    »       • 

Let  I  =  the  length  of  the  column  in  inches, 

p  =  the  least  radius  of  gyration  of  cross-section, 

I  =  the  least  moment  of  inertia  of  cross-section, 

A  =  the  area  of  the  cross-section  in  square  inches, 

Pc  =  the  breaking  load  on  the  column  in  pounds, 

/>/  =  the  mean  intensity  of  stress  under  the  breaking  load, 

or  the  unit  breaking  load,  =  Pc  -r-  A. 
pc  =  the  crushing  strength  of  the  material,  or  unit  stress 

at  the  yield  point.     This  is  the  maximum  intensity 

of  stress  in  the  column  when  the  mean  intensity  of 

stress  is  #/, 

n  =  the  factor  of  safety, 

P  =  the  working  load  on  the  column  in  pounds,  Pc  -£•  n, 
p'  =  the  mean  intensity  of  working  stress,  or  unit  working 

load,  =  p;  +  n  =  P  -5-  A, 
p  =  the    intensity   of    working    stress    in    the    column 

(  =  #c  -H  n).      This  is  the  maximum  intensity  of 

stress  in  the  column  when  the  mean  intensity  of 

stress  is  p'. 
m  —  a  coefficient   for  the  end   conditions  as  shown  in 

Table  3. 
Then  Euler's  formula  for  long  columns  is 

JEI 
P  -m—. 

v  It  is  to  be  especially  noted  that  Euler's  equation  is  rational  and 
deduced  from  the  theory  of  elasticity.  The  coefficient  m  is  also 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS 


rational  and  applicable  to  other  forms  of  column  formulae.  As 
will  be  shown  later,  the  equation  is  strictly  applicable  only  to 
very  long  columns. 

Very  short  compression  members,  of  ductile  material,  fail 
under  stresses  corresponding  to,  or  only  slightly  in  excess  of,  the 

TABLE  III 


'VALUES  OF  m  FOE.  DIFFERENT  ^ND  CONDITIONS 


CASE  I 

Fixed  at  one 

end,  free  at 

other. 


f 


CASE_H 

"Pin_Ended" 

Both  ends  free, 

but  guided. 


CASE  III 

'Pin_&  Square" 

One  end  nixed, 

the  other 

guided. 


CASE  IV 

"Square 

Ended" 

Both  ends 

fixed. 

m  ~==  4 


apparent  elastic  limit,  or  yield  point;  for  when  this  stress  is 
reached  the  metal  flows,  although  it  does  not  actually  break. 
Very  long  columns  may  approximate  the  resistance  as  given  by 
Euler's  formula.  Columns  of  lengths  intermediate  between 
compression  members  which  yield  by  simple  crushing  and  those 
which  fail  by  pure  flexure  are  weaker  than  the  former  and 
stronger  than  the  latter.  If  a  column  is  initially  exactly  straight, 
perfectly  homogeneous,  and  subjected  to  an  absolutely  concentric 
load  (that  is,  if  it  is  an  ideal  column)  there  seems  to  be  no  reason 
why  its  strength  should  diminish  rapidly  with  an  increase  of 
length,  other  conditions  remaining  the  same. 

However,  even  an  ideal  very  long  column  would  reach  the  con- 
dition of  unstable  equilibrium  when  subjected  to  a  certain  critical 
load  (the  greatest  load  consistent  with  stability).  If  the  load  is 
increased  beyond  this  limit  and  a  deflection  is  caused  in  any  way, 
the  deflection  will  increase  until  the  stress  due  to  flexure  pro- 
duces failure  of  the  column.  If  a  deflection  is  caused  while  the 
column  is  under  a  load  less  than  this  greatest  load  consistent 


64  MACHINE    DESIGN 

with  stability,  the  elasticity  of  the  material  tends  to  make  the 
column  regain  its  normal  form.  Initial  defects  in  the  form  or 
structure  of  a  column  or  eccentric  application  of  load  tend  to  pro- 
duce such  a  deflection;  hence  long  struts  fail  under  smaller  loads 
than  short  struts  of  similar  material  and  cross-section,  for  the 
ideal  conditions  are  not  realized  in  practice.  Or,  in  other  words, 
for  equal  safety  under  a  given  load  long  columns  must  have  a 
greater  cross-section,  and  lower  mean,  or  nominal,  working 
stress.*  Even  in  columns  of  moderate  length,  if  of  ductile  ma- 
terial, the  flow  at  the  yield  point  causes  buckling. 

Merriman  says  that  if  the  length  of  a  compression  member  be 
only  from  four  to  six  times  its  least  "  diameter,"  it  may  be  treated 
as  one  which  will  yield  by  simple  compression.  Johnson  gives 
limits  within  which  the  Euler  formula  should  not  be  applied  as 
/  -*-  P=  150  for  pin-ended,  and  =  200  for  square-ended  columns. 
Other  authorities  give  somewhat  different  limits;  but  nearly  all 
agree  that  most  of  the  columns  in  ordinary  structures  and  ma- 
chines are  intermediate  between  simple  compression  members 
and  those  to  which  Euler's  formulae  apply.  There  have  been  a 
great  many  column  formulae  proposed.  A  graphical  represen- 
tation of  several  of  these  formulae  is  shown  in  Fig.  13.  In  this 
diagram,  abscissas  represent  ratios  of  the  length  of  column  to 
the  least  radius  of  gyration  of  the  cross-section,  and  the  ordinates 
represent  the  nominal  (mean)  intensity  of  compressive  stress.  Or, 


x  =  l  +  p  =  l  -  v//  -5-  A,  and  y  =  p'c  =  P  -4-  A. 

The  diagram  is  drawn  for  the  ultimate  resistance  of  pin-ended 
columns  with  a  material  having  a  crushing  resistance,  pc  (yield 
point)  of  36,000  pounds  per  square  inch,  and  a  modulus  of  elas- 
ticity, E,  of  29,400,000.  The  value  of  p'e  is  36,000  for  a  very 
short  compression  member,  and  it  is  evident  that  a  long  column 
could  not  be  expected  to  have  a  greater  strength;  hence  no  for- 
mula should  be  used  which  would  give  a  value  of  pc'  in  excess  of 

*  Owing  to  the  flexure  of  the  long  column,  the  stress  is  not  uniform  across  the 
section.  The  maximum  intensity  of  stress  must  be  kept  within  the  compressive 
strength  of  the  material ;  hence  the  mean  stress  is  less  than  for  shorter  compressior 
members,  in  which  the  mean  stress  is  more  nearly  equal  to  the  maximum. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  65 

the  crushing  resistance  pc.  Referring  to  the  diagram,  it  will  ap- 
pear that  the  Euler  formula  (represented  by  the  curve  E  E,  E2) 
cannot  apply  to  pin-ended  columns  (of  this  particular  material) 
in  which  /  -H  />  <  90.  If  columns  with  a  ratio  of  /  to  />  less  than 
this  limit  yielded  by  simple  crushing,  and  those  with  a  greater 
ratio  of  Hop  followed  Euler's  formula,  the  straight  line  F  Fl  and 
the  curve  Fl  E,  E2  would  give  the  laws  for  all  lengths  of  columns. 
It  is  not  reasonable  to  expect  such  an  abrupt  change  of  law  in 
passing  this  limit  (/•*-/>  =  90) ;  and,  as  already  stated,  columns 
of  moderate  length  fail  under  a  mean  stress  considerably  less  than 
the  simple  crushing  resistance  of  the  material;  or  the  strength 


30 
20 
10 

n 

F                                   F, 

Es 
n 

35 

J-^ 

V 

^ 

^v 

X 

E\ 

r^ 

x 

\ 

N 

Vs 

Xl 

\ 

> 

£\ 

AEJ 

^. 

XH 

\\ 

RiN 

SR^ 

"^ 

^ 

x 

\J> 

N 

s 

x\ 

NX 

5 

Ss 

s^v^ 

^s 

^sj^ 

V 

r*^ 

^^v 

^^^ 

^= 

^ 

^a 

*== 

!^S 

S      — 

4 

x  = 

1-i-P 

K2 

FIG.  13. 

of  columns  is  inversely  as  some  function  of  the  length  divided 
by  the  "least  diameter." 

Mr.  Thomas  H.  Johnson  has  developed  a  formula  which  is 
based  on  the  assumption  that  the  strength  of  the  column  may  be 
taken  inversely  as  /  +  p.  This  expression  is 


(I) 


in  which  the  coefficient  k  has  the  value, 


66  MACHINE    DESIGN 

This  formula  is  represented  by  the  straight  line  T  H  J2  in  Fig. 
13.  It  will  be  noted  that  this  line  is  tangent  to  the  Euler  curve  at 
J2,  and  the  equation  of  the  latter  is  to  be  used,  should  the  columns 
exceed  the  length  corresponding  to  this  point  of  tangency 
(I  +/>  >  150)-  This  expression  is  very  simple,  after  k  has  been 
determined.  It  is  very  convenient  in  making  a  large  number  of 
computations  for  columns  of  any  one  material,  and  it  is  employed 
in  structural  work  to  a  considerable  extent.  It  does  not  appear 
to  have  any  advantage,  on  the  ground  of  simplicity,  when  some 
particular  value  of  k  does  not  apply  to  several  computations. 

For  determination  of  nominal  working  stress,  pe'  (as  computed 
above)  may  be  divided  by  a  suitable  factor  of  safety,  n.  Or  if 
p'  -r-  n  =  p',  the  expression  may  be  put  in  the  following  form  for 
direct  computation  of  mean  working  stress. 


n       n  p 

Professor  J.  B.  Johnson  has  derived  a  formula  from  the  results 
of  the  very  careful  experiments  of  Considere  and  Tetmajer. 
His  formula  is: 


for  pin-ended  columns.  The  curve  FB  Jl  (Fig.  13)  represents 
this  expression.  This  curve  is  a  parabola  tangent  to  the  Euler 
curve,  and  with  its  vertex  in  the  axis  of  ordinates  at  F,  the  direct 
crushing  stress  of  the  material.  For  columns  having  I  +  p 
greater  than  the  value  corresponding  to  the  point  of  tangency  Jl 
(should  such  be  used),  the  Euler  formula  is  to  be  employed. 
The  formula  of  Professor  Johnson's  is  empirical,  but  it  agrees 
remarkably  well  with  very  refined  experiments  on  breaking  loads. 
It  gives  considerably  higher  values  for  allowable  stress  than  other 
generally  accepted  formulas,  probably  because  it  is  based  upon 
more  refined  tests,  or  upon  conditions  further  removed  from 
those  in  practice. 

Professor  Johnson  says  ("Materials  of  Construction,"  pages 
301-302)  that  both  Bauschinger  and  Tetmajer  "mounted  their 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  67 

columns  with  cone  or  knife-edge  bearings  at  the  computed  gravity 
axis,  while  M.  Considere  mounted  his  with  lateral  screw  adjust- 
ments, and  arranged  a  very  delicate  electric  contact  at  the  side 
so  as  to  indicate  a  lateral  deflection  as  small  as  o.ooi  mm.  He 
then  applied  moderate  loads  to  the  columns  and  adjusted  the  end 
bearings  until  they  stood  under  such  loads  rigidly  vertical,  with 
no  lateral  movement  whatever."* 

It  would  appear  that  this  precaution  tends  to  make  the  test  one 
of  the  material  and  not  of  a  long  strut;  for  the  eccentricity  of 
the  load  (relative  to  the  nominal  geometric  axis)  compensates,  in  a 
measure,  for  the  lack  of  homogeneity  of  the  material.  Had  the 
correction  been  made  under  greater  load,  the  results  of  the  tests, 
if  plotted  in  Fig.  13,  would  probably  be  still  nearer  the  line  F  Flt 
and  the  difference  between  these  test  columns  and  columns  as 
used  in  practice  would  be  greater,  requiring  a  higher  contingency 
factor  in  the  latter  for  safety. 

For  determining  the  working  stress,  the  value  of  pc'  (as  com- 
puted from  the  above  form  of  Johnson's  expression)  should  be 
divided  by  a  suitable  factor  of  safety  n.  Or,  the  formula  may  be 
put  in  the  following  form  for  computing  nominal  working  stress: 

.....  « 


The  Rankine  or  Gordon  formula  (see  Church's  "  Mechanics," 
pages  372-376)  has  been  extensively  used  for  columns.  It  may 
be  expressed  as  follows  : 


The  above  formula  is  based  upon  experiments  on  the  breaking 
strength  of  columns.  The  coefficient  P  is  purely  empirical,  and 
this  fact  limits  its  usefulness,  for  it  leaves  much  uncertainty  as 
to  how  this  coefficient  should  be  modified  for  materials  different 
from  those  which  have  been  actually  tested  as  columns.  The 

*  "  This  precaution  is  essential  to  a  perfect  test  of  the  material.  .  .  .  Only  in 
this  way  can  other  sources  of  weakness  be  eliminated." — [J.  B.  J.] 


68  MACHINE    DESIGN 

mean  intensity  of  working  stress,  p> ',  might  be  inferred  by  divid- 
ing pe'  by  n,  or  the  expression  can  be  written : 

(6) 


I 

I  +  m 

but  it  is  not  entirely  satisfactory  to  assume  the  action  for  stresses 
within  the  elastic  limit,  from  the  results  of  tests  for  breaking 
strength.  The  form  of  the  Rankine  expression  is  rational,  but 
the  coefficient  ft  is  not. 

Professor  Merriman  says,  in  his  "  Mechanics  of  Materials," 
page  (129) :  "Several  attempts  have  been  made  to  establish  a  for- 
mula for  columns  which  shall  be  theoretically  correct.  .  .  .  The 
most  successful  attempt  is  that  of  Ritter,  who,  in  1873,  proposed 
the  formula 


i  +  - 

"The  form  of  this  formula  is  the  same  as  that  of  Rankine's 
formula,  .  .  .  but  it  deserves  a  special  name  because  it  com- 
pletes the  deduction  of  the  latter  formula  by  finding  for  ft  a  value 
which  is  closely  correct  when  the  stress  p  does  not  exceed  the 
elastic  limit  pc."  The  above  notation  is  changed  to  agree  with 
that  previously  used  in  this  article.  The  ratio  pe  -r-  p  is  the  factor 
of  safety.  For  ultimate  strength,  this  formula  might  be  written: 
P  t> 

t/  c  £  c  /AT"    \ 


but  the  first  form  (eq.  N)  is  the  more  important.  The  curve 
R!  T  R2  (Fig.  13)  is  the  graphical  representation  of  the  last 
expression,  eq.  N1* 

Merriman  gives  the  Euler  formula  for  a  factor  of  safety  of  n  = 
Pc  +  P)  which  is 

D  J*  /  „  \  2 

....     (9) 


*  Professor  Merriman  developed  equation  Ni  independently,  but  later  than 
Ritter.  He  gives  Ritter  sole  credit  for  the  formula  in  the  1897  edition  .of  his 
"Mechanics  of  Materials." 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  69 

Failure  occurs  if  p  >_  pc.  The  Ritter  formula  (eq.  N)  reduces 
to  this  last  expression  for  columns  so  long  that  the  term  unity 
in  the  denominator  is  negligible;  strictly  speaking,  this  is  only 
the  case  when  /  H-  />  =  infinity.  Professor  Merriman  also  shows, 
mathematically,  that  the  two  curves,  E  El  E2  and  Rl  T  R2,  are 
tangent  to  each  other  when  I  +  p  =  infinity. 

If  /^-jo  =  0,  the  Ritter  formula  reduces  to  p'=P+A,  which 
is  the  ordinary  formula  for  short  compression  members. 

The  facts  that  this  formula  is  rational  in  form,  that  it  gives  the 
correct  values  at  the  limits  /  --  p  =  oo  and  /  -f-  />  =  o,  and  that  it 
lies  wholly  within  the  boundary  F  Fl  E^  E2  (Fig.  13),  all  justify  its 
use,  and  it  will  be  adopted  in  this  work.  It  will  be  noted  from 
Fig.  13  that  the  Ritter  and  Rankine  formulas  agree  very  closely 
for  the  material  taken  for  illustration ;  but  the  fact  that  the 
curve  of  the  latter  crosses  the  Euler  curve  near  the  right-hand 
limit  of  the  diagram  indicates  that  its  constant  /?  is  not  theo- 
retically correct. 

Exception  may  be  taken  to  the  use  of  the  Ritter  formula  for 
cast  iron,  since  it  involves  the  use  of  the  stress  at  the  elastic 
limit,  and  the  coefficient  of  elasticity,  both  of  which  have  no 
definite  fixed  values  for  cast  iron.  But  the  same  criticism  ap- 
plies to  the  use  of  any  rational  formula  founded  on  the  elastic 
theory,  as  far  as  cast  iron  is  concerned.  Thus  the  expres* 
sions  for  deflection  in  simple  beams  contain  E  which,  for  cast 
iron,  may  vary  from  15,000,000  to  20,000,000.  Since  cast-iron 
columns  designed  simply  for  strength  are  very  rare  in  machine 
design  it  therefore  seems  best  to  use  the  formula  since  otherwise 
it  fulfils  all  needs  better  than  any  other. 

If  it  is  desired  to  design  a  cast-iron  column  with  great  accuracy 

values  of  — ~-=  may  be  taken  which  will  give  results  in  ac- 
m  TT    rL 

cordance  with  experiment  and  which  practically  transforms  the 
equation  into  Rankine's  formula.  If  — j^-=  =  q,  then  for  cast- 
iron  columns  with  fixed  ends  q  =  -  — ,  for  one  end  fixed  and  the 

5,000' 


7° 


MACHINE    DESIGN 


I.78 


other  free  but  guided  q  = ,  and  for  both  ends  free  but  guided 

q  =  -     — .     In  addition  the  student  should  consult  treatises  on 


5,000 
the  strength  of  materials  treating  fully  of  this  subject. 

All  of  the  above  formulas  give  the  value  of  the  mean  ultimate 
stress  (p'  =  Pc  +  A),  or  the  mean  working  stress  (p'  =  P  +  A), 
corresponding  to  a  maximum  ultimate  stress  pc  or  a  maximum 
working  stress  p,  respectively.  However,  the  ordinary  problem 


Stress  In  1000  Lbs. 

6  8  10 


0       10      20 


140 


of  design  is  to  assign  proper  dimensions  for  the  member  under 
the  given  load.  It  is  not  practicable  to  solve  directly,  for  the 
area  in  such  expressions  as  those  given  in  this  article  as  p'  (or  p) 
and  p  are  both  functions  of  the  area  of  the  cross-section.  It  is 
usual  to  assume  a  section  somewhat  larger  than  that  demanded 
for  simple  crushing,  and  then  to  check  for  the  ultimate  load  P, 
or  the  working  load  P' '.  Professor  W.  N.  Barnard  has  devised 
a  diagram  which  is  very  convenient  for  these  computations  for 
steel  or  wrought-iron  columns.  It  is  shown,  to  a  reduced  scale, 
in  Fig.  13  (a).  The  four  curves  are  for  the  four  end  conditions 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  71 

given  in  Table  III,  page  63.  They  are  plotted  for  a  maximum 
working  stress  of  10,000  pounds  per  square  inch,  and  a  value  of 

d,  =  -  which  is  an  average  value  for  steel.     The  curves 

E       29,500,000 

should  not,  however,  be  used  for  cast  iron,  wood,  or  other  mate- 
rials where  the  ratio  ~  will  give  values  far  different  from  the 

EL 

above,  but  such  cases  may  be  solved  directly  by  equation  N. 
They  may  be  used  for  any  other  stress  by  proceeding  as  follows: 
Assume  a  trial  cross-section,  which  fixes  />.  Divide  /  by  this 
value  of  p]  take  this  quotient  on  the  lower  scale  and  pass  directly 
upward  to  the  proper  curve  for  the  given  end  conditions;  then 
pass  horizontally  to  that  one  of  the  radiating  diagonals  which  is 
numbered  to  correspond  with  the  selected  stress;  from  this  last 
point  pass  upward  to  the  horizontal  scale  at  the  top  of  the  diagram, 
where  the  value  of  the  unit  load  or  mean  working  stress  (/>')  is 
read  off.*  If  this  value  of  pf  agrees  sufficiently  well  with  the 
quotient  of  the  load  divided  by  the  trial  area,  the  section  may  be 
considered  as  satisfactory. 

In  the  case  of  a  square-ended  column,  or  when  the  supporting 
action  of  the  ends  is  equal  in  all  possible  planes  of  flexure,  it  is 
sufficient  to  take  the  least  radius  of  gyration  of  the  section;  or 
to  take  p  for  the  axis  about  which  the  section  is  weakest.  In 
case  of  a  pin-ended  column,  as  a  connecting-rod,  the  cylindrical 
supporting  pins  make  it  equivalent  to  a  square-ended  column 
against  flexure  in  the  plane  of  the  axes  of  the  pins,  provided  these 
bear  symmetrically  with  reference  to  the  axis  of  the  column ;  while 
the  column  is  pin-ended  with  reference  to  a  plane  perpendicular 
to  the  axes  of  the  pins.  If  the  cross-section  of  such  a  column 
has  equal  dimensions  in  these  two  planes  (circular,  square  sec- 
tions, etc.),  the  column  need  only  be  computed  for  the  latter 
plane.  If  the  pin-ended  column  has  an  oblong  section  (elliptical, 
rectangular  but  not  square,  I  section,  etc.),  it  may  be  weaker  in 


*  The  method  of  using  the  diagram  is  indicated  by  the  arrows,  for  an  example 
in  which  l-^-p  =  8o  and  the  maximum  working  stress=  14,000  (pin-ended).  In  this 
case,  p'  is  found  to  be  about  7,900. 


72  MACHINE    DESIGN 

either  of  these  two  planes,  notwithstanding  the  difference 
in  end  conditions  relative  to  them;  and  it  may  be  neces- 
sary to  compute  for  both  planes,  unless  the  section  is  ob- 
viously stronger  in  one  of  them.  If  a  rectangular,  or  elliptical, 
column  has  a  section  in  which  the  dimension  in  the  plane 
of  the  pins  is  more  than  one-half  the  dimension  in  the 
plane  perpendicular  to  the  pins,  it  will  suffice  to  compute  as 
a  pin-ended  column  against  flexure  in  the  latter  plane,  and 
vice  versa. 

In  the  preceding  discussion,  the  various  formulae  have  been 
given  both  for  breaking  and  for  working  loads.  The  Euler  and 
Ritter  formulas  are  derived  from  the  theory  of  elasticity;  hence 
these  are  proper  for  computations  pertaining  to  working  loads,  in 
which  the  stress  should  never  exceed  the  elastic  limit.*  It 
does  not  follow  that  these  two  rational  formulas  will  agree  with 
experiments  on  the  ultimate  resistance  of  columns  or  for  materials 
which  do  not  follow  Hooke's  law  of  proportionality  of  stress  to 
strain.  These  expressions  are,  in  this  respect,  like  the  common 
beam  formulae.  Such  formulae  as  Rankine's  and  J.  B.  Johnson's, 
derived  from  tests  of  ultimate  resistance  of  columns,  are,  for 
similar  reasons,  less  rigidly  applicable  to  working  loads  and 
stresses. 

Example.  The  connecting-rod  of  a  steam  engine  is  5 
feet  long  and  is  subjected  to  a  load  of  20,000  Ibs.  If  the 
maximum  allowable  stress  is  9,000  Ibs.  per  sq.  in.,  deter- 
mine the  diameter  of  a  circular  section  at  the  centre  of  the 
rod.  Take  £  =  30,000,000,  and  the  elastic  limit  £,  =  36,000 
Ibs.  per  sq.  in.  The  rod  may  be  considered  a  pin-ended 
column.  Hence  m  =  i. 

If  the  rod  were  designed  as  a  short  column,  the  required  area 

would  be  A  =  —    —  =  2.250.   ins.    or    a   diameter    of   i44- 
9,000 

inches;    and  it  is  evident  that  for  a  long  column  the  diameter 
must  be  greater  than  this.     Assume  2K  inches  as  a  trial  diameter. 

*  The  Euler  formula  is  not  applicable  for  practical  applications,  except  for 
quite  long  columns. 


9,000 

T           I 

36,000  ^ 

60 

1  ix, 

r2  X  3O,OOO,OOO 

5 

T 

STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  73 

d         t 

Then  A  =  4-9?  P  *=-  =  ^  /  =  60'',  whence  in  N 
4       o 


p' 


/.  P  =  4,300X4.9  =  2 1, 070  Ibs.  which  is  a  little  more  than  the 
required  load  and  the  section  will  fulfil  the  requirements.  The 
student  should  also  follow  the  solution  through  on  the  diagram. 
21.  Eccentric  Loading  of  Long  Columns.  In  the  preced- 
ing discussion  of  columns  it  has  been  assumed  that  the  load  has 
been  applied  axially.  This  is  obviously  the  best  way  of  applying 
the  load,  but  cases  often  occur  where  it  must  be  applied  at  a  dis- 
tance a  from  the  axis  of  the  column.  In  such  a  case  the  column 
is  said  to  carry  an  eccentric  load,  and  the  arm  a  is  called  the 
eccentricity.  If  the  length  of  the  column  be  less  than  4  or  6 

times  its  least  diameter,  that  is,  if  the  ratio  —  be  less  than  about 

25,  the  member  may  be  treated  by  the  method  outlined  in  para- 

P       Pae 

graph  (10)  and  formula  M  will  apply  or  p  =—  +  — - — . 

A.  J. 

If,  however,  the  column  be  longer  than  4  to  6  times  its  least 

P 
diameter,  it  can  no  longer  be  assumed  that  the  direct  stress  — 

due  to  the  load  is  uniformly  distributed  over  the  section,  as  it 
has  been  shown  by  the  discussion  on  long  columns  that  such  is 
not  the  case. 

In  addition,  if  the  load  is  applied  eccentrically,  it  is  obvious 
that  the  column  will  deflect  somewhat  more  than  it  would  if  the 
load  were  applied  axially.  This  will  have  the  effect  of  adding  to 
the  original  lever  arm  a  an  additional  amount  «,  due  to  this  de- 
flection. 

The  stresses  therefore  acting  on  an  eccentrically  loaded 
column  are — 

(a)  A  compressive  stress  plt  such  as  would  be  induced  if  the 
load  were  axial. 


74  MACHINE    DESIGN 

(b)  A  flexural  stress  p2,  due  to  the  eccentricity  and  propor 
tional  to  the  bending  moment  P  (#  +  «). 
For  the  first  from  Ritter's  formula  (N) 

'' 


and  for  the  second  from  (/) 

P  (a  +  «)  e       P  (a  +  «)  e 

~T~  ^O^ 

Therefore  the  maximum  compressive  stress*  in  the  section  is 


For  columns  whose  ratio   of  —  is  less  than  100,  and  working 

stresses  such  as  must  be  used  in  machine  design,  the  deflection 
«  may  be  neglected.  For  columns  longer  than  this,  or  where  the 
stress  is  necessarily  high,  «  can  be  determined  by  the  theory  of 
elasticity.  For  a  full  discussion  of  the  manner  of  computation 
see  Merriman's  "Mechanics  of  Materials,"  1905  edition,  page 
217.  For  the  ordinary  cases  of  machine  design  this  refinement 
may  be  omitted. 

Example.  A  circular  wooden  pole  30  feet  high  is  required 
to  carry  a  transformer  weighing  800  pounds,  with  an  eccentricity 
of  10  inches.  What  must  be  the  diameter  at  the  middle  in  order 
that  the  stress  due  to  this  load  shall  not  exceed  500  pounds  per 
square  inch?  Let  pe  =  3,000  pounds  per  square  inch  and  E  = 

1,500,000.     Also  m  =  -.     (See  Table  III.) 
4 

*  The  stress  induced  on  the  convex  side  of  an  eccentrically  loaded  column  may 
be  either  tensile  or  compressive,  but  will  always  be  less  than  the  stress  on  the  con- 
cave side.  For  materials  whose  elastic  strength  is  about  the  same  in  either  tension 
or  compression  the  stress  on  the  convex  side  is  of  no  importance.  If,  however,  the 
column  is  made  of  a  material,  such  as  cast  iron,  whose  tensile  strength  is  much  less 
than  its  compressive  strength,  the  character  and  magnitude  of  the  stress  on  the 
convex  side  should  be  investigated.  If  e'  be  the  distance  from  the  neutral 
axis  to  the  outer  fibre  on  the  convex  side,  then  the  stress  (p)  on  the  convex  side  is, 

I£  *ls  P°sitive  the  stress  is 


sile;    if  p  is  negative  the  stress  is  compressive. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  75 

Assumed  a  diameter  of  8".     Then  />  =  2  and  A  =  50 


Whence  p  =  «^[  ,  +  -  ^o  -  ,360  y     10X4-1 
50  L          ^X  7T2  X  1,500,000  V   2   /  4    J 

4 

800  F 

—  [i  +  26  + 

=  592  pounds  per  square  inch. 

If  this  excess  is  considered  too  great,  a  second  approximation 
can  be  made. 

22.  Stress  Due  to  Change  of  Temperature.  Practically  all 
metals  expand  when  heated,  and  contract  again  when  cooled. 
The  amount  which  a  bar  expands  per  unit  of  length,  for  a  rise 
of  one  degree  in  temperature,  is  called  its  coefficient  of  linear 
expansion,  and  will  be  denoted  by  C.  The  following  table  gives 
values  of  C  for  various  substances  for  one  degree  Fahrenheit  : 

Hard  Steel   ..........  C  =  .0000074 

Soft  Steel   ...........  C  =  .0000065 

Cast  Iron   ...........  C  =  .0000062 

Wrought  Iron  ........  C  =  .0000068 

If  a  bar  of  metal  is  held  at  the  ends,  so  as  to  prevent  it  from 
expanding  or  contracting,  stresses  are  produced  in  it  which  are 
called  temperature  stresses  ;  the  effect  being  the  same  as  though 
the  bar  had  been  compressed,  or  elongated,  an  amount  corre- 
sponding to  its  expansion  or  contraction  due  to  the  change  in 
temperature. 

Let  t  =  change  in  temperature  in  degrees. 

Let  p  =  stress  induced  per  unit  area. 

stress        p  r    _, 

Since  E  =  -  —  =  ^-  .    .  p  =  C  t  h 
strain       C  t 

Example.  A  bar  of  wrought  iron  2"  square  is  raised  to  a 
temperature  of  100  degrees  above  its  normal.  If  held  so  that  it 
cannot  expand,  what  stress  will  be  induced  in  it,  and  what  force 
must  oppose  it  to  prevent  expansion  ? 


76  MACHINE    DESIGN 

Let  E  =  30,000,000 

p  =  C  t  E  =  .0000068  X  100  X  30,000,000  =  20,400  Ibs. 
and  the  total  opposing  force  P  will  be 

P  =  20,400  X  4  =  81,600  Ibs. 

23.  Resilience.  In  all  the  previous  discussions  on  the  vari- 
ous straining  actions  to  which  a  member  may  be  subjected,  it 
has  been  assumed  that  the  load  was  a  simple  dead  load  and 
applied  without  initial  velocity  or  impulse.  But,  as  already 
pointed  out,  the  load  may  be  applied  impulsively;  or  it  may  be 
applied  in  any  way,  and  removed  and  applied  again  and  again 
repeatedly.  The  application  of  a  load  in  an  impulsive  manner, 
or  the  repeated  application  of  a  load,  does  not  affect  the  charac- 
ter of  the  straining  action,  but  does  affect  the  amount  of  stress  or 
strain.  In  order  to  more  clearly  discuss  the  effect  of  impulsive 
loading  it  will  be  necessary  to  consider  the  straining  effect  of  a 
load  somewhat  more  fully;  the  discussion  of  repeated  loads  will 
be  given  in  a  succeeding  section. 

If  a  material  is  distorted  by  a  straining  action,  it  is  capable  of 
doing  a  certain  amount  of  work  as  it  recovers  its  original  form. 
If  the  deformation  does  not  exceed  the  elastic  strain,  this  amount 
of  work  is  equal  to  the  work  done  upon  the  material  in  producing 
such  deformation.  If  the  material  is  strained  beyond  the  elastic 
limit,  it  returns  work  only  equal  to  that  expended  in  producing 
elastic  deformation;  and  the  energy  required  to  cause  the  plastic 
deformation,  or  set,  is  not  recovered,  as  it  is  not  stored  but  has 
been  expended  in  producing  such  permanent  change  of  form. 
Ordinary  springs  illustrate  the  first  case;  the  shaping  of  ductile 
metals  by  forging,  rolling,  wire-drawing,  etc.,  are  processes  in 
which  nearly  all  of  the  energy  is  expended  in  producing  perma- 
nent deformation. 

The  work  required  to  produce  a  strain  in  a  member  is  called 
the  work  of  deformation.  If  the  strain  produced  is  equal  to 
the  deformation  at  the  true  elastic  limit,  the  energy  expended  is 
called  elastic  resilience*  If  the  piece  is  ruptured,  the  energy 

*  When  the  term  resilience  is  used  without  qualifying  context,  elastic  resilience 
is  to  be  understood. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  77 

expended  in  breaking  it  is  called  total  work  of  deformation. 

If  O  a  d  e  (Fig.  6)  is  the  stress-strain  diagram  for  a  given  mate- 
rial, the  area  O  a  a'  represents  the  elastic  resilience,  and  O  a  d  e  ef 
represents  the  total  work  of  deformation  per  cubic  inch  of  the 
material. 

In  such  materials  as  have  well-marked  elastic  limits  (propor- 
tionality between  stress  and  strain  through  a  definite  range)  the 
line  Oa  is  a  sensibly  straight  line,  and  the  elastic  resilience 
Oaa!  =  y^aaf  X Oaf;  or,  the  elastic  resilience  equals  the  elastic  strain 
(Oa'}  multiplied  by  one-half  the  elastic  stress  (/4  aa') .  The  area 
Oadee'  equals  the  base  (Oef)  multiplied  by  the  mean  ordinate  (y) 
of  the  curve  Oade;  or,  if  the  quotient  of  this  mean  ordinate  of 
the  curve  divided  by  the  maximum  ordinate  be  called  k,  the 
work  of  deformation  equals  the  ultimate  strain  multiplied  by  k 
times  the  maximum  stress.  It  is  evident  that  for  a  straining 
action  beyond  the  elastic  limit,  k  >  K  and  k  <  i. 

The  curve  OADEE'  represents  the  stress-strain  diagram  of  a 
material  having  higher  elastic  and  ultimate  strength  than  the 
former.  The  greater  inclination  of  the  elastic  line  (OA)  with 
the  axis  of  strain  (OX)  shows,  in  the  second  case,  a  higher  modulus 
of  elasticity,  as  this  modulus  equals  the  elastic  stress  divided 

by  the  elastic  strain.     In  the  first  case  El  =  -—  in   the   sec- 

A  A' 
ond  case,  E2  =  -^7. 

The  stress-strain  diagram  OADEE'  shows  that  of  two  mate- 
rials one  may  have  both  the  higher  elastic  and  ultimate  strength, 
and  still  have  less  elastic  resilience  and  less  total  work  of  deforma- 
tion. If  the  curve  O  a"  d"  e"  is  the  stress-strain  diagram  of  a 
third  material  (having  a  modulus  of  elasticity  similar  to  the 
first),  it  appears  that  this  third  material  possesses  greater  elastic 
resilience,  but  less  total  work  of  deformation  than  the  first. 

A  comparison  of  these  illustrative  stress-strain  diagrams  (for 
quite  different  materials)  also  shows  that,  for  a  given  stress,  the 
more  ductile,  less  rigid  material  may  have  the  greater  resilience. 
Hence,  when  a  member  must  absorb  considerable  energy,  as  in 


78  MACHINE    DESIGN 

case  of  severe  shock,  a  comparatively  weak  yielding  material  may 
be  safer  than  a  stronger,  stiffer  material.  This  is  frequently 
recognized  in  drawing  specifications.  The  principle  is  similar  to 
that  involved  in  the  use  of  springs  to  avoid  undue  stress  from 
shock.  In  fact  springs  differ  from  the  so-called  rigid  members 
only  in  the  degree  of  distortions  under  loads,  or  in  having  much 
greater  resilience  for  a  given  maximum  load. 

If  a  material  is  strained  beyond  its  elastic  limit,  as  to  a'  (Fig. 
13  b),  upon  removal  of  the  load  it  will  be  found  to  have  such  a 
permanent  set  as  O  O' '.  Upon  again  applying  load,  its  elastic 
curve  will  be  O'  a';  but  beyond  the  point  a'  its  stress-strain  dia- 
gram will  fall  in  with  the  curve  which  would  have  been  pro- 
duced by  continuing  the  first  test  (i.e.,  a'de).  Similarly,  if 
loaded  to  a",  the  permanent  set  is  O  O",  and  upon  again  apply- 
ing load,  the  stress-strain  diagram  becomes  O"  a"  d  e.  The 
elastic  limit  a"  of  the  overstrained  material  is  evidently  higher 
than  the  original  elastic  limit,  a;  while  the  original  total  work  of 
deformation,  O  a  d  e,  is  considerably  greater  than  the  total  work 
of  deformation  of  the  overstrained  material,  O"  a"  d  e.  The 
effects  of  strain  beyond  the  elastic  limit  are  thus  seen  to  be : 

I.  Elevation  of  the  elastic  strength  and  increase  of  the  elastic 
resilience. 

II.  Reduction  of  the  total  work  of  deformation. 

These  facts  have  an  important  influence  on  resistance  to  re- 
peated shock.  The  above  noted  elevation  of  the  elastic  limit  by 
overstraining  can  usually  be  largely  or  wholly  removed  by 
annealing. 

24.  Suddenly  applied  Load,  Impact,  Shock.  It  will  perhaps 
be  well  to  first  consider  the  general  case  of  a  load  impinging  on 
the  member,  with  an  initial  velocity;  this  velocity  (v)  correspond- 
ing to  a  free  fall  through  the  height  h.  For  simplicity,  the  dis- 
cussion will  be  confined  to  a  load  producing  a  tensile  stress;  but 
the  formulae  will  apply  equally  well  to  uniform  compressive  and 
shearing  stresses,  and  all  except  (5)  apply  directly  to  cases  of 
torsion  and  flexure. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS 


79 


W  =  static  value  of  load  applied  to  member. 

h  =  height   corresponding   to   velocity   with   which   load   is 
applied. 

8  =  total  distortion  of  member  due  to  impulsive  load. 

p  =  maximum  intensify  of  resulting  stress. 
A  =  area  of  cross-section  of  the  member. 
P  =  p  A  =  total  max.  stress  due  to  load  as  applied  suddenly. 

A  =  total  distortion  of  member  due  to  static  load,  W. 
x  =  h  +  k  (for  convenience). 

£  =  a  constant;   its  value  is  /4  if  E.  L.  is  not  passed;  but  if 
E.  L.  is  exceeded  k  >   %  and  k  <  i. 


O' 


O' 


O'" 


FIG.  13  (c). 


FIG.  13  (b). 


The  energy  to  be  absorbed  by  the  member  due  to  the  impulsive 
application  of  the  load  is  W  (h  +  $)',  the  work  of  deformation  is 
k  Pd.  (See  preceding  article,  Resilience.) 

Case  I.  —  Maximum  Stress  within  Elastic  Limit. 


rr. •••;*-£ 


(2) 

(3) 


80  MACHINE    DESIGN 

_ 

.        (4) 


-  _ 
p  =  -     =  -     (i  +  V  i  +  2  *)      ;  -  .     .-  .     .          (5) 


5  =        =  A  (i  +  V  i  +  2  *)        .      .      .  ...      ..    .     (6) 

The  elongation  at  the  elastic  limit  equals  F+E,  in  which  E  = 
modulus  of  elasticity  and  F  =  intensity  of  stress  at  the  elastic 
limit. 

If  L  =  length  of  the  member, 

(i  +  L):(F  +  E):  :  (W  -5-  A)  :F;  .'.  A  =  WL  +  A  E.  .  (7) 
As  A  is  small  for  metals  (except  in  the  forms  of  springs)  a 
moderate  impinging  velocity  may  produce  very  severe  stress.  It 
will  be  evident  that  A  and  <S  are  directly  proportional  to  the  length 
of  the  member;  hence  the  stress  produced  by  a  given  velocity  of 
impact  (height  h)  is  reduced  by  using  as  long  a  member  as  pos- 
sible. 

If  the  load  is  applied  instantaneously,  but  without  initial  veloc- 
ity, h  =  Q  and  #  =  o;  whence 

p-w(i  +  V7T5)  =  2  w    .    .....    (30 

#  =  ^  =  —  (50 

p      A         A          ..........     15; 


S  =  A  (i   +  V  !  +  0)  =  2  A (60 

Case  II. — Maximum  Stress  Beyond  the  Elastic  Limit.  If  the 
maximum  stress  exceeds  the  elastic  limit,  the  constant  k  of  equa- 
tion (i)  is  between  %  and  i  (see  Art.  23,  Resilience),  and  its  exact 
value  cannot  be  determined  in  the  absence  of  the  stress-strain  dia- 
gram for  the  particular  material.  Thus  (Fig.  13  c),  W  (h  +  8), 
is  represented  by  the  rectangle  m  n  c  q;  and  this  area  must  equal 
the  area  O  a  b  c;  the  latter  being  greater  than  the  elastic  resili- 
ence, O  a  a',  and  less  than  the  total  work  of  deformation  O  a  d  e  er , 
in  this  illustration. 

When  the  stress-strain  diagram  is  known,  the  following  prob- 
lems can  be  readily  solved : — 

(a)  Determination  of  the  velocity  of  impinging  of  a  given  load 
(or  corresponding  value  of  h)  to  produce  a  given  stress,  or  strain. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  8 1 

(b)  Determination  of  the  load  which  will  produce  any  par- 
ticular stress,  or  strain,  when  impinging  with  a  given  velocity. 

(c)  Determination  of  the  stress,  or  strain,  produced  by  a  given 
load  impinging  with  a  given  velocity. 

Let  the  work  of  deformation  corresponding  to  the  known 
stress,  or  strain,  in  (a)  and  (b) ,  be  called  R  =  k  P  d.  If  the 
stress-strain  diagram  is  for  stress  per  unit  of  sectional  area  and 
strain  per  unit  of  length  of  the  member,  let  W  be  the  load  per 
unit  of  sectional  area;  hf  the  height  due  the  velocity  of  impinging 
divided  by  the  total  acting  length  of  the  member;  8'  the  distor- 
tion per  unit  of  length  of  the  member  due  to  impulsive  load;  and 
R'  the  resilience  for  unit  of  volume,  or  the  modulus  of  resilience. 


(a)  :  W  (h' 

+  *')  =  k  j. 

>  V  =  Rf. 

.'  .  h' 

W 

•     ,     -     (7) 

Rf 

(b}  :  W  =  - 

f8) 

(c) :  The  solution  of  this  problem  is  not  quite  so  definite,  in 
the  general  case,  as  the  preceding;  but  it  can  be  easily  accom- 
plished, graphically,  with  sufficient  accuracy.  Draw  the  line  g  q 
(Fig.  13  c)  (indefinitely),  parallel  to  O  ef ,  and  at  a  distance  from 
it  equal  to  W'\  take  out  the  area/*g  =  gOt.  Whatever  the 
value  of  3',  the  shaded  area  O  c  qfig  O  =  W  tf\  hence  the  un- 
shaded area  under  the  stress-strain  curve  must  equal  W  hf.  A 
few  trials  will  suffice  to  locate  the  limiting  line  b  q  c  which  will 
give/*'  b  qf  =  m  n  O  t  =  W  hf. 

The  case  in  which  the  maximum  stress  is  within  the  elastic 
limit  is  by  far  the  most  important,  as  it  is  almost  always  desired 
to  keep  the  maximum  intensity  of  stress,  P  +  A,  within  the 
elastic  limit,  especially  as  every  overstrain  (beyond  this  limit) 
raises  the  elastic  limit  and  decreases  the  total  resilience  (see  Fig. 
13).  The  effect  of  a  shock  which  strains  a  member  beyond  the 
elastic  limit  is  to  reduce  its  margin  of  safety  for  subsequent 
similar  loads,  because  of  reduction  in  its  ultimate  resilience. 
Numerous  successive  reductions  of  the  total  resilience  by  such 
actions  may  finally  cause  the  member  to  break  under  a  load 
which  it  has  often  previously  sustained. 
6 


82  MACHINE    DESIGN 

No  doubt  many  cases  of  failure  can  be  accounted  for  by  the 
effects  just  discussed;  but  there  is  another  and  quite  different 
kind  of  deterioration  of  material,  which  is  treated  in  the  follow- 
ing article. 

Dr.  Thurston  has  shown  that  the  prolonged  application  of  a 
dead  load  may  produce  rupture,  in  time,  with  an  intensity  of 
stress  considerably  below  the  ordinary  static  ultimate  strength 
but  above  the  elastic  stress.  It  is  well  known  that  an  apprecia- 
ble time  is  necessary  for  a  ductile  metal  to  flow,  as  it  does  flow 
when  its  section  is  changed  under  stress;  hence,  a  test  piece  will 
show  greater  apparent  strength  by  quickly  applying  the  load 
than  by  applying  it  more  slowly,  provided  the  application  of 
load  is  not  so  rapid  as  to  become  impulsive. 

The  kind  of  failure  which  is  the  subject  of  the  next  topic  is  due 
to  a  real  permanent  deterioration  of  the  metal,  and  it  is  due  to 
distinctly  different  causes  from  those  mentioned  above. 

25.  On  the  Peculiar  Action  of  Live  Load.  Fatigue  of  Metals. 
It  has  been  found  by  experience  and  experiment,  that  materials 
which  are  subjected  to  continuous  variation  of  load  cannot  be 
depended  upon  to  resist  as  great  stress  as  they  will  carry  if  applied 
but  once,  or  only  a  few  times.  When  the  load  is  suddenly  applied, 
and  frequently  repeated,  the  decline  of  strength  or  of  the  power  of 
endurance  may  perhaps  be  ascribed,  in  part  at  least,  to  the  eleva- 
tion of  the  elastic  limit  and  reduction  of  the  ultimate  resilience,  as 
discussed  in  Art.  24.  But  apart  from  this  cause,  with  repeated 
loads,  even  in  the  absence  of  appreciable  shock,  a  decided  de- 
terioration of  the  material  very  frequently  occurs.  This  effect 
has  been  called  the  Fatigue  of  Materials,  although  some  authorities 
restrict  this  term  to  the  kind  of  deterioration  already  referred  to  as 
the  simple  result  of  a  decrease  of  resilience.  The  term  fatigue 
implies  a  weakening  of  the  material  due  to  a  general  change  of 
structure.  It  was  formerly  supposed  that  the  repeated  variation 
of  stress  caused  such  change  of  the  general  structure,  possibly 
owing  to  slight  departure  from  perfect  elasticity  under  stress 
much  below  that  ordinarily  designated  as  the  elastic  limit.  The 
crystalline  appearance  of  the  fracture  sustained  this  view;  but 
numerous  tests  of  pieces  from  a  member  ruptured  in  this  way, 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  83 

(taken  as  near  as  possible  to  the  break) ,  fail  to  show  such  crystalline 
fracture,  and  it  is  difficult  to  reconcile  the  normal  appearance  and 
behavior  of  such  test  pieces  with  the  theory  of  general  change  of 
structure. 

A  theory  which  has  been  largely  accepted  is  that  every  piece 
of  metal  contains  innumerable  minute  flaws  or  imperfections, 
often  originally  too  small  to  be  detected  by  ordinary  means.  These 
"micro-flaws"  tend  to  extend  across  the  section  under  variation 
of  stress,  and  may,  in  time,  reduce  the  net  sound  section  so  greatly 
that  the  intensity  of  stress  in  the  fibres  which  remain  intact  be- 
comes equal  to  the  normal  breaking  strength  of  the  material. 
Professor  Johnson  suggests:  "the  gradual  fracture  of  metals"  as 
a  more  appropriate  term  than  "  fatigue."  Many  men  of  large  prac- 
tical experience  still  prefer  wrought  iron  to  mild  steel  for  various 
members  which  are  subject  to  constantly  reversing  stress. 

It  is  probable  that  the  prejudice  against  steel  is  largely  the 
result  of  unskilful  manipulation  of  this  more  sensitive  mate- 
rial; and  the  product  of  the  best  steel  makers  of .  to-day  is  much 
stronger  and  more  reliable  than  wrought  iron. 

However,  it  is  just  possible  that  the  very  lack  of  homogeneity 
in  wrought  iron  renders  it  safer  under  varying  stress  (other  things 
being  equal) ,  as  the  fibres  are  more  or  less  separated  by  the  streaks 
of  slag,  and  a  flaw  is  less  apt  to  extend  across  the  entire  section 
than  it  is  in  the  continuous  structure  of  steel.  Wrought  iron  may 
be  likened  to  a  wire  rope,  in  which  a  fracture  in  one  wire  does 
not  directly  extend  to  adjacent  wires. 

The  " gradual  fracture"  through  extension  of  " micro-flaws  " 
seems  to  accord  with  the  observed  facts  more  closely  than  the 
older  theory  of  general  change  of  structure. 

In  the  American  Machinist  (Sept.  27,  1906)  will  be  found  an 
account  of  recent  researches  tending  to  show  that  metals  are 
made  up  of  grains,  each  grain  consisting  of  many  crystals,  and 
that  when  deformation  takes  place  in  a  metal  these  crystals  move 
relatively  to  each  other  along  "gliding  planes."  If  the  stress 
producing  such  sliding  is  repeated  often  enough  the  contact  at 
the  gliding  planes  weakens  and  finally  passes  into  a  crack  or 
series  of  cracks  which  extend  across  the  section. 


84  MACHINE    DESIGN 

The  theory  of  the  subject  is,  as  yet,  too  incomplete  to  permit 
of  derivation  of  rational  formulae  to  account  for  the  effects  of  re- 
peated live  loads;  and  if  the  "micro-flaw"  theory  is  correct,  it 
is  not  probable  that  such  rational  analysis  can  ever  be  satisfac- 
torily applied. 

All  of  the  formulae  that  have  been  derived  for  computation  of 
breaking  strength  under  known  variations  of  load,  or  stress,  are 
empirical  ones  which  have  been  adjusted  to  fit  the  experiment- 
ally determined  facts. 

Consult:  Johnson's  "Materials  of  Construction." 
Merriman's  "  Mechanics  of  Materials." 
Unwin's  "Testing  of  Materials." 
Weyrauch    (Du    Bois):  "  Structure   of    Iron    and 
Steel." 

Experiment  has  shown  that  the  breaking  strength  under  re- 
peated loading,  or  the  "  carrying  strength,"  is  a  function  of  the 
magnitude  of  the  variation  of  stress  and  of  the  number  of  repeti- 
tions of  such  varying  stress.  Furthermore,  this  function  is 
different  for  different  materials;  and  there  are  authentic  observa- 
tions on  record  which  go  to  show  that,  as  between  different  mate- 
rials, the  one  with  the  higher  static  breaking  strength  does  not 
always  possess  the  greater  endurance  under  repeated  loading.  In 
general,  however,  the  carrying  strength  under  repeated  loads  is  a 
function  of  the  static  strength. 

The  allowable  working  stress  usually  depends  upon :  (a)  The 
number  of  applications  of  the  load.  This  should  be  considered 
as  indefinite,  or  practically  infinite,  in  many  machine  members, 
(b)  The  range  of  load.  This  is  frequently  either  from  zero  to 
a  maximum;  or  between  equal  plus  and  minus  values,  (c)  The 
static  breaking  strength  or  the  elastic  strength. 

The  first  systematic  experiments  upon  the  effect  of  repeated 
loading  were  conducted  by  Wohler  [1859  to  1870].  He  found, 
for  example,  that  a  bar  of  wrought  iron,  subjected  to  ten- 
sile stress  varying  from  zero  to  the  maximum,  was  ruptured 
by: 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS 


800  repetitions  from  o  to  52,800  Ibs.  per  sq.  in. 
107,000         "  "      o  to  48,000 

450,000  "      o  to  39,000  " 

10,140,000  "      o  to  35,000        "          u 

— Merriman,  page  191. 

It  was  found  that  the  stress  could  be  varied  from  zero  up  to 
something  less  than  the  elastic  limit  an  indefinite  number  of 
times  (several  millions)  before  rupture  occurred;  but  with  com- 
plete reversal  of  stress,  or  alternate  equal  and  opposite  stresses, 
(tension  and  compression),  it  could  be  broken,  by  a  sufficient 


20000 


60000 


40000   xx 


Tension 


Compression. 


"T 
i 


i 

.1 


FIG.  13  (d). 

number  of  applications,  when  the  maximum  stress  was  only 
about  one-half  to  two-thirds  the  stress  at  the  elastic  limit. 

A  number  of  efforts  have  been  made  to  deduce  from  the  exper- 
iments of  Wb'hler,  formulae  which  could  be  applied  to  the  design 
of  machine  members  (see  Unwin,  page  36).  One  of  the  best  of 
these  formulae  is  that  of  Professor  Johnson  as  it  is  easily  applied 
to  all  cases  that  will  arise;  it  is  simpler  than  most  of  those  previ- 
ously proposed;  and  it  is  probably  as  reliable  as  any  yet  offered. 

Two  formulae  which  have  been  very  generally  accepted  for 
computing  the  probable  carrying  strength  are:  Launhardt's  for 


86  MACHINE    DESIGN 

varying  stress  of  one  kind  only,  and  Weyrauch's  for  stress  which 
changes  sign. 

Suppose  a  material  to  have  a  static  ultimate  strength  u  of 
60,000  Ibs.  per  sq.  in.  If  the  minimum  unit  strength  be  plotted 
as  a  straight  line,  A  O  B  (Fig.  13  d),  the  locus  of  the  maximum 
unit  stress,  from  the  Launhardt  formula,  is  the  broken  curve 
from  B  to  D.  That  is,  for  example,  when  the  minimum  tensile 
stress  is  12,500,  the  maximum  tensile  carrying  stress  would  be 
about  40,000;  or  the  material  could  be  expected  to  stand  an 
indefinite  number  of  loadings  if  the  range  of  stress  did  not 
exceed  15,000  to  40,000  pounds  per  square  inch  in  tension.  In  a 
similar  way,  the  broken  curve  from  D  to  C  is  the  locus  of  maxi- 
mum tension,  from  the  Weyrauch  formula,  when  the  locus  of 
minimum  stress  (negative  tension,  or  compression)  is  the  straight 
line  A  O.  It  will  appear  that  the  straight  line  C  D  B  agrees 
fairly  well  with  these  two  curves.  Inasmuch  as  it  seems  un- 
reasonable to  expect  an  abrupt  change  of  law  when  the  minimum 
stress  passes  through  zero,  and  as  there  is  no  rational  basis  for 
the  Launhardt  and  Weyrauch  formulae,  it  appears  reasonable  to 
adopt  the  upper  straight  line  as  the  locus  of  the  maximum  stress. 
Owing  to  the  discrepancies  in  the  observations  (which  must  be 
expected  from  the  probable  cause  of  the  deterioration  of  the 
metal),  this  straight  line  may  be  accepted  as  representing  the 
law  as  accurately  as  could  be  expected  of  any  empirical  line. 
These  are,  in  substance,  the  reasons  given  by  Professor  Johnson 
for  basing  his  formula  on  the  straight  line  C  D  B.  For  full  dis- 
cussion and  derivation  of  the  following  formula,  see  Johnson's 
"Materials  of  Construction,"  pages  545-547. 

Let  p2  =  maximum  intensity  of  stress. 
/>!  =  minimum  intensity  of  stress. 
u  =  ultimate  (static)  intensity  of  stress. 

Then  in  general  : 


As  the  expressions  contain  the  ratio  of  the  minimum  to  maxi- 
mum intensities  of  stress,  instead  of  their  difference,  they  are  ap- 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  87 

plicable  when  the  area  of  cross-section  of  the  member  is  unknown; 
for  whatever  this  area,  the  ratio  of  the  stresses  is  the  same  as  the 
ratio  of  the  loads  producing  these  stresses.  In  substituting  values 
of  pi  and  />2,  care  must  be  taken  to  use  proper  signs;  thus,  if 
tension  is  taken  as  positive,  compression  is  negative;  or,  if  the 
stress  varies  between  tension  and  compression  p2  is  positive  and 
pl  is  negative. 

For  dead  load,  p^  =  pz  ; 


P2 

For  repeated  load  when  pl  =  o,  —  =  o 

P2 


For  complete  reversal  of  load,  pl  =  —  p2 


The  three  special  cases  (2),  (3),  and  (4),  are  those  most  com- 
monly met  with  in  designing,  but  the  general  expression  (i) 
should  not  be  lost  sight  of. 

Example.  A  bar  of  steel,  whose  ultimate  static  tensile  strength 
is  70,000  Ibs.  per  sq.  inch,  is  subjected  to  a  repeated  load  whose 
minimum  value  is  one  half  the  maximum  value.  What  is  the 
maximum  stress  that  can  be  carried  by  the  bar  for  an  indefinite 
number  of  repetitions? 

Since  the  stress  will  be  proportional   to  the  load  pl  =  —  - 
Hence  substituting  in  equation  (i),  p2  =  —  -  =  47,000. 


It  is  to  be  noted  that  the  allowable  maximum  stress  is  above  the 
original  elastic  limit  of  most  steel,  and  if  the  piece  were  designed 
to  be  stressed  to  47,000  Ibs.  the  result  would  be  that  the  first 
application  of  the  load  would  raise  the  elastic  limit  to  that  value. 


88  MACHINE    DESIGN 

But  the  piece  would  take  permanent  set  and  be  in  most  cases  of 
no  further  use.  A  factor  of  safety  must  therefore  be  used  in 
order  that  the  maximum  stress  may  be  well  below  the  elastic 
limit. 

The  experiments  of  Wohler,  and  his  successor  in  the  field, 
Baushinger,  were  conducted  on  a  very  limited  variety  of  mate- 
rials; so  that  while  the  above  discussion  points  out  what  may  be 
expected  in  a  general  way  from  most  materials,  they  are  not  suf- 
ficiently conclusive  to  make  it  possible  to  pick  out  the  exact 
factor  of  safety  to  be  used  in  all  cases.  They  do,  however,  throw 
much  light  on  the  apparently  high  factors  of  safety  which  must 
sometimes  be  used,  and  for  which  no  other  satisfactory  explana- 
tion has  been  found. 

26.  The  Factor  of  Safety.  The  preceding  paragraphs  (arti- 
cles 9  to  26)  have  considered  the  effect  that  different  methods 
of  applying  the  load  will  have  on  a  member,  and  the  relations 
which  exist  between  a  given  dead  load  and  the  resulting  stress 
and  strain.  It  has  been  shown  in  Art.  24  that  if  the  load  is 
applied  suddenly  the  resulting  stress  and  strain  will  be  twice  as 
great  as  for  a  dead  load.  And  finally  in  Art.  25  it  has  been 
shown  that  the  maximum  stress  that  can  with  safety  be  induced 
repeatedly  in  a  member,  will  depend  on  the  range  of  stress.  It 
would  seem  as  though  a  member  designed  in  accordance  with 
these  logical  theories  would  be  satisfactory.  But  it  must  be 
remembered  that  these  theories  are  not  absolute,  that  the  in- 
formation regarding  the  characteristics  of  materials  is  still  very 
incomplete;  that  flaws  and  hidden  defects  always  exist;  and 
finally  that  there  is  always  danger  of  accidental  overloading. 

In  addition,  it  is  generally  essential  that  a  machine  member  be 
not  only  strong  enough  to  avoid  breaking  under  the  regular 
maximum  working  load,  but  also  that  it  shall  not  receive  a 
permanent  set;  for  a  machine  member  ordinarily  becomes  useless 
if  it  takes  such  set  after  it  has  been  given  the  required  form.  In 
many  cases  a  temporary  strain,  even  considerably  below  that 
corresponding  to  the  elastic  limit,  would  seriously  impair  the 
accuracy  of  operation;  and  in  such  cases  the  member  often  re- 
quires great  excess  of  strength  to  secure  sufficient  rigidity.  It 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS  89 

follows,  therefore,  from  these  considerations  that  if  the  design  of 
a  machine  member  were  based  on  the  maximum  allowable  stress, 
as  indicated  by  Wohler's  experiments  (such  stress  being  modi- 
fied by  the  theory  of  suddenly  applied  loading,  should  it  be 
present),  there  would  be  no  margin  to  allow  for  the  uncer- 
tainties and  unknown  defects  enumerated  above;  and  in  many 
cases  leave  no  assurance  that  the  elastic  limit  would  not  be  ex- 
ceeded. So  that  while  stresses  fixed  in  accordance  with  these 
theories  form  a  good  basis,  they  must  in  general  be  reduced  by 
means  of  a  factor  of  safety  so  that  the  working  stress  is  enough 
lower  to  provide  for  these  uncertainties. 

The  factor  of  safety  is  generally  defined  as  the  quotient  of  the 
ultimate  static  strength  divided  by  the  working  stress.  A  con- 
sideration of  Wohler's  experiments  shows  that  such  a  definition 
is  misleading.  For  a  factor  of  safety  of  2,  for  instance,  might 
be  perfectly  safe  for  a  dead  load;  but  for  a  repeated  load  with 
stress  in  one  direction  it  would  leave  no  margin  at  all  for  contin- 
gencies. The  apparent  factor  of  safety  would  seem  to  be  a 
better  term,  and  the  real  factor  of  safety  may  be  defined  as  the 
quotient  of  the  carrying  strength,  or  maximum  allowable  stress 
as  given  by  Wohler's  experiments,  divided  by  the  working  stress. 

The  factor  of  safety  has  been  called  the  "  factor  of  ignorance," 
and,  as  it  is  too  often  applied,  it  is  perhaps  little  else.  Thus 
very  often  it  is  specified  that  all  the  members  of  a  machine  shall 
be  designed  with  a  certain  fixed  factor  of  safety  without  regard 
to  the  conditions  under  which  the  various  members  may  have  to 
act.  A  factor  of  safety  applied  in  this  manner  is,  generally 
speaking,  a  factor  of  ignorance.  It  is  probable  that  the  factor 
of  safety  will  always  retain  an  element  of  ignorance,  for  it  can 
hardly  be  hoped  that  the  powers  of  analysis  will  ever  permit  the 
prediction  of  the  exact  effect  of  every  possible  straining  action, 
due  to  regular  service  and  accident.  Neither  can  it  be  expected 
that  the  methods  of  manufacture,  and  inspection,  will  become  so 
perfect  as  to  eliminate  or  measure  precisely  every  possible  defect 
in  materials  or  workmanship.  But  a  careful  study  of  the  condi- 
tions of  each  particular  case  and  a  proper  attention  to  the  effects 
which  may  be  weighed  (at  least  approximately)  should,  with 


QO  MACHINE    DESIGN 

the  knowledge  now  to  be  had,  enable  the  designer  to  make  a 
fairly  accurate  application  of  the  factor  of  safety,  an  intelligent 
choice  of  which  is  the  most  important  part  of  design. 

Most  of  the  formulae  of  Mechanics  which  are  applicable  to  the 
design  of  machine  members,  are  based  on  theoretical  treatment 
of  the  stresses  induced  by  the  action  of  given  forces  within  the 
elastic  limit  upon  the  member  under  consideration;  and  the 
theoretical  conclusions  so  reached  are  amply  verified  by  practical 
experiment.  When,  therefore,  the  conditions  under  which  the 
member  is  to  work  can  be  analyzed,  and  the  laws  of  Mechanics 
applied  to  its  design,  such  methods  as  outlined  in  this  chapter 
are  perfectly  rational,  if  intelligent  allowance  is  made  for  contin- 
gencies. Many  machine  members,  however,  are  subjected  to 
such  a  complicated  system  of  stress  that  analysis  cannot  be 
strictly  applied,  and  less  satisfactory  approximations  or  assump- 
tions are  unavoidable  in  the  present  state  of  knowledge.  When 
such  is  the  case,  the  designer  must  either  base  the  design  on  the 
predominating  stress,  if  there  is  such,  allowing  such  a  margin  or 
factor  of  safety  as  experience  or  experiment  may  show,  to  pro- 
vide for  the  minor  uncertain  stresses;  or,  if  the  case  considered 
be  beyond  such  treatment,  recourse  must  be  had  to  empirical 
methods  or  judgment.  (See  Art.  i.) 

While  therefore  mathematical  treatment  of  any  case  will  serve 
as  a  good  guide  to  correct  proportions,  such  treatment  must 
always  be  tempered  with  judgment,  a  high  development  of 
which  is  necessary  to  successful  design,  as  in  all  other  branches 
of  engineering. 

While,  also,  no  fixed  rules  for  selecting  the  factor  of 
safety  can  be  laid  down,  a  knowledge  of  Wohler's  experiments, 
and  the  effect  of  suddenly  applied  loads,  will  greatly  aid  the 
designer  in  the  matter.  Thus  when  it  is  known  that  the  load  is 
to  be  a  dead  load,  an  apparent  factor  of  safety  of  3  will,  for 
wrought  iron,  or  steel,  bring  the  working  stress  well  below  the 
elastic  limit  and  allow  something  for  contingencies.  If,  however, 
the  load  be  a  repeated  load,  the  stress  varying  from  zero  to  a 
maximum  tensile  stress,  the  apparent  factor  of  safety  for  steel 
must  at  least  be  5,  to  allow  a  good  margin  below  the  elastic  limit; 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS 


and  in  either  case,  if  in  addition  the  load  is  to  be  suddenly  applied, 
these  factors  must  be  multiplied  by  2  to  insure  safety. 

Example.  A  steel  beam  is  subjected  to  a  suddenly  applied 
load  which  alternately  induces  an  equal  tensile  and  compressive 
stress;  if  the  ultimate  strength  be  60,000  Ibs.  per  sq.  in.,  what 
apparent  factor  of  safety  should  be  used,  and  what  will  be  the 
real  factor  of  safety? 

Since  the  stress  is  a  reversed  one,  the  maximum  allowable 
stress  or  carrying  strength  is  by  Wohler's  experiments  one-third 
of  ultimate  strength  or  20,000  Ibs.  If  the  working  stress  is  one- 
half  of  this  value  or  10,000  Ibs.,  it  will  leave  a  good  margin  for 
contingencies,  disregarding  the  impulsive  effect.  But  the  load 
is  applied  suddenly  and,  by  Art.  24,  this  value  (10,000)  must 
be  again  divided  by  2,  making  the  working  stress  5,000  Ibs.  per 

sq.  in.     Therefore  the  apparent  factor  of  safety  is  -     —  =  12, 

5,000 

while  the  real  factor  based  on  Wohler's  law  is  — =  4. 

5,000 

If  the  member  should  have  to  work  under  extremely  trying 
conditions,  or  if  shock  or  other  stresses  which  could  not  be 
analyzed  were  present,  this  value  might  have  to  be  still  further 
reduced. 

TABLE  IV 

FACTORS    OF   SAFETY 


CHARACTER  OF 
MATERIAL. 

DEAD 
LOAD. 

REPEATED 
STRESS  IN  ONE  DIRECTION. 

REPEATED 
REVERSED  STRESS. 

£* 

^.S-d 

¥ 

£t3 

s^-s 
•SHI 

•33$ 
co^ 

1-2 

'rt.JJ'd 

P 

£"C 

111 

w 

Wrought  Iron,  Steel, 
or    other    Ductile 
Metals 

3 
4 

5 
6 

10 
12 

6 
10 

12 

2O 

Cast  Iron,  or    other 
Brittle  Metals  

Table  4  contains  factors  of  safety  such  as  are  used  in  practice 
and  which  agree  fairly  well  with  the  foregoing  theory.  They 
are,  of  course,  average  values  and  must  be  used  with  judgment ; 


92  MACHINE    DESIGN 

but  in  the  absence  of  trained  judgment,  or  as  an  aid  to  its  de- 
velopment, they  may  be  found  useful. 

Table  5  contains  values  of  the  ultimate  strengths  and  elastic 
limits  of  the  materials  most  used  in  engineering.  They,  also,  are 
average  values  such  as  the  designer  must  use  in  the  absence  of 
exact  information  regarding  the  material  to  be  employed,  and  in 
general  such  exact  information  is  lacking. 

It  may  be  observed  that  an  increased  factor  of  safety  may  not 
always  in  the  case  of  cast  metals  give  a  stronger  member.  If 
the  increased  dimensions  give  sections  so  thick  that  sponginess 
results,  the  gain  in  strength  may  be  negative;  and  when  internal 
pressure,  such  as  is  found  in  hydraulic  work,  is  to  be  withstood, 
it  is  often  necessary  to  do  with  a  smaller  factor  of  safety  to 
insure  soundness. 


STRAINING    ACTIONS    IN    MACHINE    ELEMENTS 


93 


I 

w 

g 

>  g 


M      W 

2  I 


Jf4 


O     O     O     O     O     O 


94 


MACHINE    DESIGN 
TABLE  VI 


CHARACTER  OP  STRESS 
OR  STRAIN. 


FORMULA. 


A         Stress  in  Ten.  or  Comp 

B         Strain  in  Ten.  or  Comp 

C         Stress  in  Shear 

D         Torsional  Stress 

E         Torsional   Stress,    Solid    Cir- 
cular Shaft 

F          Torsional  Stress,  Hollow  Cir- 
cular Shaft 

G         Torsional    Strain,  Solid   Cir- 
cular Shaft 

H         Torsional  Strain,  Hollow  Cir- 
cular Shaft 

I  Deflection  in  Bending 
J  Stress  due  to  Flexure 
K  Combined  Bend'g  and  Twist'g 

£L 

:E* 

£a 

L         Combined  Torsion  and  Com- 
pression 

LI        Combined  Torsion  and  Com- 
pression 

M        Combined  Flexure  and  Direct 

Stress 

N        Long  Column 

O         Eccentric    Loading    of   Long 
Columns 


P== 


PI 

AE 


P*  =  -T 


Pa  =  T  = 


psl 


Pa  =  T 


16 


0  = 


32  Tl 


TtEs(dS—dS} 


See  Table  I. 


M  =  tL     See  Table  I. 


Te  =  V  M2  +  T2 


P  + 


64  rn 


>c        (l\\    (a  +  a)e 

^E(-P)  +-^~ 


STRAINING  ACTIONS  IN  MACHINE  ELEMENTS 


95 


TABLE  VII 

PROPERTIES    OF    SECTIONS 


Shape 

of 
Section. 


Moment 

of 
Inertia. 

I 


Modulus 

of 
Section. 


Square  of 
Eadius  of 
Gyration. 


Polar 

Moment  ofi 

Inertia, 

J 


D2 


32 


£M 


32 


BTT3 

12 


_H2 
12 


12 


-bh° 


-bh 


1  r"BH3-bh3] 
12  [EH      bh  J 


k-B-*; 


^[BH3—  .bh3J 


(BH2-bh2)  -4  BH  bh  (H-h)2 
12(BH— bh) 


I_(BH2-bh2)-4BHbh(H-h)2 
ei~"  6(BH2-bh2) 

1  _^(BH2-bh2)— 4BH  bh  (H-h)2 
es     "      6(BH2-2bhH+bh.2) 


bH3-fB»h8 

12(bH-|-Bh) 


BH3 
36 


T        RH2 
I  =."" 

6!        24, 

I      BH2 

e2~  12 


,H2 
18 


" 


64 


irBH* 
32 


7r(BH34-HB3 
64, 


CHAPTER  IV 

GENERAL    THEORY    OF    FRICTION,    LUBRICATION,    AND 

EFFICIENCY 

27.  Friction  in  General.  When  two  solid  surfaces  are  held  in 
contact  by  any  appreciable  force,  any  effort  tending  to  move  them 
relatively  to  each  other  is  met  by  a  resisting  force  acting  tangen- 
tially  to  the  surface  of  separation  of  the  two  bodies.  This  resist- 
ance to  relative  motion  is  due  to  the  interlocking  of  the  minute  de- 
pressions and  elevations  which  exist  even  in  the  smoothest  surfaces 
and  will,  of  course,  vary  with  different  properties  of  materials 
and  different  qualities  of  finish.  Thus,  unsurfaced  cast  iron  will 
show  a  very  great  resistance  to  relative  motion,  while  two  hard- 
ened and  ground  surfaces  of  steel  will  move  over  each  other 
with  much  more  ease.  If  the  two  surfaces  are  very  carefully 
fitted  together  without  any  foreign  matter  in  between,  they  will, 
in  the  case  of  many  substances,  adhere  firmly  together,  which 
still  further  increases  the  resistance  to  relative  motion.  If  oils 
or  lubricants  of  any  kind  are  interposed  between  the  surfaces, 
the  resistance  to  relative  motion  is,  to  a  considerable  extent, 
overcome. 

This  tendency  to  resist  relative  motion  is  sometimes  a  desirable 
feature  and  sometimes  not.  In  the  case  of  bearing  and  rubbing 
surfaces  generally,  such  frictional  resistances  result  in  loss  of 
power  and  should  be  reduced  to  a  minimum;  while  in  the  case 
of  friction  clutches,  brake  straps,  keys,  screw  fastenings,  etc., 
frictional  resistance  is  of  great  utility  and  every  effort  is  made 
to  insure  its  presence.  The  laws  of  friction,  and  the  manner 
of  their  application  therefore,  are  of  prime  importance  to  the 
engineer. 

These  laws  are  at  present  rather  imperfectly  understood, 
though  considerable  experimental  work  has  been  done.  It 
has  been  found  that  many  of  the  older  theories  based  on  ex- 

96 


GENERAL    THEORY    OF    FRICTION 


97 


perimental  work  are  true  only  for  the  range  of  conditions 
covered  by  the  experiments,  and  that  conditions  different  from 
these  show  entirely  different  results. 

The  ratio  of  frictional  resistance  F,  to  the  normal  load  P,  is 
called  the  coefficient  of  friction;  or  if  this  ratio  be  denoted 

F 

by/,  then,  for  flat  surfaces,  f  =  —  or  F  =  fP. 

In  the  case  of  circular  surfaces,  such  as  journals  and  bearings, 
the  distribution  of  the  normal  pressure  is  variable  and  depend- 
ent on  the  manner  in  which  the  surfaces  are  fitted  together. 
In  such  cases  it  is  customary  for  convenience  to  define  the  co- 
efficient of  friction  in  a  similar  way  as  in  flat  surfaces,  and  con- 
sider that  it  has  special  values  for  circular  surfaces.  The  co- 
efficient of  friction  for  circular  surfaces  will  be  denoted  by  p. 
Hence  as  before,  F  =  ft  P. 

The  intensify  of  normal  pressure  on  circular  surfaces,  as  before 
stated,  is  difficult  of  accurate  determination  and  it  is  therefore 
customary  to  take  as  the  normal  pressure  the  intensity  of  pres- 
sure per  unit  of  projected  area.  Or,  if  d  =  diameter  of  shaft, 
and  /  =  length  of  bearing,  then  the  intensity  of  pressure  per 

P 

unit  of  projected  area,  w  =  —.  . 

U>  If 

The  energy  absorbed  by  frictional  resistance  is  transformed 
into  heat  which  is  conducted  away  by  conduction  and  radiation 
to  the  air,  or,  in  the  case  of  certain  kinds  of  bearings,  by  water 
circulation  or  other  means.  The  work  of  friction  is  often  there- 
fore an  important  factor  in  the  design  of  rubbing  surfaces.  For 
flat  plates  the  foot  pounds  of  energy  absorbed  per  minute  is 
E  =fPV,  where  V  the  velocity  is  in  feet  per  minute  and  P  is 
in  pounds.  For  circular  surfaces,  if  N  be  the  number  of  revolu- 
tions per  minute,  and  d  the  diameter  of  shaft  in  inches, 


12 


*For  other  forms  of  surfaces  see  Kent's  "Engineer's  Pocketbook,"  page  938, 
and  Thurston's  "  Friction  and  Lost  Work,"  page  40. 
7 


Qo  MACHINE    DESIGN 

If  then /or  /*  be  known,  for  any  pair  of  rubbing  surfaces,  the 
frictional  resistance  and  the  energy  absorbed  for  any  load  P  may 
be  calculated.  Values  of /and  p  have  been  obtained  experimen- 
tally for  many  of  the  materials  and  conditions  met  with  in  engi- 
neering, but  the  data  so  far  available  are  still  incomplete. 

The  consideration  of  the  laws  of  friction,  as  applied  to  ma- 
chinery, naturally  divides  itself  into  two  parts. 

(a)  Friction  of  Dry  or  Unlubricated  Surfaces. 

(b)  Friction  of  Lubricated  Surfaces. 

28.  Friction  of  Unlubricated  Surfaces.  The  experiments 
of  Morin,  Rennie,  Coulomb,*  and  many  others,  furnish  the  fol- 
lowing laws  for  dry  or  very  slightly  lubricated  surfaces. 

(1)  The  frictional  resistance  is  approximately  proportional  to 
the  normal  load. 

(2)  The  frictional  resistance  is  approximately  independent  of 
the  extent  of  the  surfaces. 

(3)  The  frictional  resistance,  except  at  very  low  speeds,  de- 
creases as  the  velocity  increases. 

It  was  formerly  supposed  that  an  abrupt  change  took  place  in 
the  value  of  /  when  the  body  passed  from  a  state  of  motion  to 
one  of  rest.  It  seems  now,  however,  that  while  the  coefficient 
of  rest  is  in  general  greater  than  that  of  motion,  the  change  in 
value  is  gradual  and  the  value  at  rest  is  not  far  different  from 
that  at  very  slow  motion.  As  the  velocity  increases,  the  value 
of  /  materially  decreases  and  this  must  be  taken  account  of 
in  designing  machinery  where  friction  is  involved.  Unfortu- 
nately the  information  regarding  high  or  even  moderate  speeds  is 
also  very  incomplete. 

The  following  values  of/  must,  in  view  of  the  incomplete  infor- 
mation, and  also  because  of  variations  which  come  with  slight 
changes  of  conditions,  be  looked  on  as  approximate  values  only. 
Unless  it  is  positively  known  that  the  surfaces  will  be  kept  free 
from  even  slight  contamination  by  oily  substances,  these  values 
must  be  used  with  judgment. 

*  See  "Lubrication  and  Lubricants":  Archbutt  &  Deely,  for  a  full  discussion 
of  these  points. 


GENERAL    THEORY   OF   FRICTION  99 

Coefficients  of  Friction  (f)for  Dry  or  Slightly  Lubricated 
Surfaces. 

Wood  on  Wood — Static  or  very  low  velocity 3  to  .5 

Wood  on  Metals       "       "       "     "         "          2  to  .6 

Leather  on  Metals    "       "       "     "         "          3  to  .6 

Leather  on  Wood     "       "       "     "         "          3  to  .5 

Metal  on  Metal        "      "        "     "        "         (average)     .3 
Cast  Iron  on  Steel — velocity  =    440  feet  per  minute ...     .32 

"          "          "          "  "          =    2640        "        "  "         .  .  .        .2 

"         "          "          "  "  =   5280        "        "  "         .  .  .        .06 

There  are  no  experimental  data  giving  the  decrease  in  the 
value  of/  at  high  speeds,  for  combinations  such  as  wood  or  leather 
on  metals.  The  data  for  cast  iron  on  steel  will,  however,  serve  as 
a  rough  guide  to  what  may  be  expected  to  occur.  It  is  to  be  par- 
ticularly noted  that,  in  designing  brake  shoes  or  other  friction 
machinery  where  great  velocities  are  involved,  allowance  must 
be  made  for  the  decrease  in  the  value  of  the  coefficient. 

29.  Dry  Rolling  Friction.     When  a  curved  body  rolls  upon 
a  plane  or  curved  surface,  it  has  been  found  that  the  so-called 
frictional  resistance  due  to  the  rolling  action  is  much  less  than 
that  due  to  sliding,  for  the  same  load.     If  P  =  the  load;  F  =  the 
horizontal  force  required  at  the  axis  of  a  circular  body  to  pro- 
duce and  sustain  uniform  motion;  and  r  =  radius  of  rolling  body, 

kP 

it  has  been  found  that  F  = where  k  is  a  coefficient  to  be  de- 

r 

termined  experimentally.  If  r  be  expressed  in  inches  k  is  found 
to  have  a  value  of  about  .02  for  iron  or  steel  rolling  on  iron  or 
steel. 

Neither  the  coefficient  k  nor  the  exact  theory  of  rolling  friction 
is  at  present  very  accurately  known.  The  most  important  use 
of  rolling  friction  is,  as  far  as  the  present  discussion  is  concerned, 
in  connection  with  roller  bearings  for  shafting,  and  a  fuller  dis- 
cussion of  these  will  be  given  later. 

30.  Friction  of  Lubricated  Surfaces.     When  a  lubricant  is 
interposed  between  a  pair  of  rubbing  surfaces,  the  frictional 
resistance  is  materially  reduced  because  the  surfaces  are  wholly  or 


100  MACHINE    DESIGN 

partially  separated  from  each  other  by  the  lubricant.  The 
lubricant  may  be  fed  to  the  surfaces  in  a  number  of  ways.  If  the 
motion  is  intermittent,  and  other  conditions  will  allow,  a  simple 
oil  hole  leading  to  the  rubbing  surfaces  is  often  used.  If  the 
motion  is  continuous,  some  form  of  oil  cup  which  will  give  a  con- 
tinuous supply  is  better.  Fig.  14  (a)  shows  a  cup  of  the 
simpler  type  where  a  wick  of  cotton  or  wool  draws  up  the  oil  by 
capillary  attraction  and  feeds  it  slowly  into  the  oil  hole.  This 
is  sometimes  called  siphon  feed.  Fig.  14  (b)  shows  a  so- 
called  sight  feed  cup  where  the  oil  falling  by  gravity  from  the 
cup  can  be  seen  as  it  passes  the  hole  e  and  the  flow  can  be  regu- 
lated by  the  screw  d.  Centrifugal  action  is  also  used  to  some 
extent  to  feed  oil  to  rotating  parts.  Sometimes  an  opening  is 
made  in  the  bearing  so  that  a  pad  saturated  with  lubricant  can  be 
kept  pressed  up  against  the  moving  surface,  thus  lubricating  the 
whole  length  of  the  journal  continuously.  For  heavy  lubricants, 
such  as  greases,  where  very  heavy  pressures  are  carried  on  the 
rubbing  surfaces,  so-called  compression  cups  are  often  used  and 
are  constructed  so  as  to  force  the  lubricant  in  between  the  sur- 
faces. Fig.  14  (c)  shows  a  "ring  oiled"  bearing.  The  ring  r 
running  loose  on  the  shaft  s  dips  into  the  pocket  below  the  shaft. 
The  friction  of  the  ring  on  the  shaft  causes  it  to  rotate  and  draw 
up  oil  from  the  pocket.  Sometimes  chains  are  used  instead  of 
solid  rings.  For  the  most  efficient  lubrication  the  journal  itself 
runs  in  a  bath  of  oil  (Fig.  15)  or  is  flooded  with  oil  supplied  under 
pressure.  The  relative  merits  of  these  various  methods  of  sup- 
plying the  lubricant  will  be  more  apparent  after  a  discussion  of 
the  general  laws  of  lubrication. 

The  effect  of  friction,  and  the  efficiency  of  lubrication  of  so- 
called  lubricated  surfaces,  may  conveniently  be  treated  under 
three  heads : — 

(a)  Static  Friction  of  Lubricated  Surfaces. 

(b)  Friction  of  Imperfectly  Lubricated  Surfaces. 

(c)  Friction  of  Perfectly  Lubricated  Surfaces. 

31.  Static  Friction  and  Lubrication.  When  a  pair  of  lu- 
bricated surfaces  are  pressed  together  by  a  load,  the  pressure 
tends  to  slowly  expel  the  lubricant  from  between  the  surfaces. 


GENERAL   THEORY  OF   FRICTION 


101 


Experiments  and  experience  show  that  it  is  very  difficult  even 
with  limited  areas  and  heavy  pressures  completely  to  expel  the 
lubricant.  If  ordinary  machinery,  however,  is  allowed  to  stand 
at  rest  for  a  short  period  of  time,  this  action  is  sufficient  to  expel 
so  much  of  the  lubricant  that  may  have  been  between  the  sur- 
faces while  running  as  to  allow  the  metallic  surfaces  to  come 
more  or  less  in  contact.  The  static  coefficient  of  friction  of  lubri- 
cated surfaces  is  hence  very  much  higher  than  that  of  surfaces 
whicn  move  even  very  slowly;  for  it  will  be  seen  presently  that  even 
at  low  velocities  the  surfaces  tend  to  draw  in  the  lubricant  by  their 
motion.  It  is  a  well-known  fact  that  heavy  machinery  always 
ofters  a  great  resistance  to  starting  after  lying  idle  a  short  time 
and  often  the  rubbing  surfaces,  if  not  oiled  before  starting,  will 


FIG.  15. 


abrade  each  other  before  the  lubricating  action  due  to  running 
begins  to  take  effect.  The  materials  therefore  for  the  rubbing 
surfaces  of  heavy  machinery  should  be  carefully  chosen  for  their 
antifriction  qualities,  and  oil  grooves  should  be  carefully  provided 
so  that  lubricant  can  be  applied  as  near  the  point  of  greatest 
pressure  as  possible  before  motion  begins. 

The  coefficient  of  static  friction  for  lubricated  surfaces  is  not 
very  accurately  known  and  it  varies  somewhat  with  the  pres- 
sure and  character  of  the  lubricant.  A  fair  average  value  for 
metal  surfaces  and  pressures  ranging  from  75  to  500  Ibs.  per  sq. 
in.  is  .15.* 


*See  Thurston's  "Friction  and  Lost  Work,"  pages  316-317. 


102  MACHINE    DESIGN 

32.  Imperfect  Lubrication.  When  one  lubricated  surface 
slides  over  another,  the  moving  surface,  even  at  low  velocities, 
tends  to  carry  the  lubricant,  if  properly  applied,  in  between  the 
surfaces.  Thus  the  layer  of  oil  which  touches  the  surface  of  a 
journal  adheres  to  it  and  is  carried  along  under  the  bearing. 
This  layer  in  turn  tends  to  carry  along  the  layer  which  next  ad- 
joins it,  because  the  viscosity  of  the  lubricant  opposes  the  shear- 
ing action  which  results  between  layers  on  account  of  the  action 
of  the  moving  surface  of  the  journal.  In  plane  sliding  surfaces 
the  lubricant  is  generally  applied  to  the  stationary  surface  and 
tends  to  cling  to  it  in  spite  of  the  tendency  of  the  slider  to  rub 
it  off.  The  action  of  the  sliding  surfaces  in  drawing  in  the  lu- 
bricant is  similar  to  that  of  the  rotating  journal,  but  in  a  much 
less  marked  degree  as  would  naturally  be  expected  from  the  na- 
ture of  the  case.  If  the  velocity  of  rubbing  be  very  low,  or  the 
pressure  very  high,  or  the  supply  of  lubricant  limited,  the  quantity 
of  lubricant  that  is  carried  in  is  very  small  and  the  surfaces  in 
contact  are  very  slightly  lubricated  and  may  even  be  in  actual 
metallic  contact.  The  materials,  therefore,  for  the  rubbing  sur- 
faces of  slow-moving  machinery  should  also  be  carefully  chosen 
for  their  antifriction  qualities,  as  even  after  the  machinery  has 
been  successfully  set  in  motion  metallic  contact  may  occur 
between  them. 

If  the  velocity  of  rubbing  and  the  supply  of  lubricant  be  in- 
creased, the  load  remaining  the  same,  more  and  more  lubricant 
is  thrust  between  the  surfaces  by  the  action  noted  above  till,  at 
a  point  depending  on  the  pressure,  velocity  of  rubbing,  and  vis- 
cosity of  the  lubricant,  the  metallic  surfaces  are  completely  sep- 
arated and  the  friction  becomes  only  that  due  to  the  fluid  fric- 
tion of  the  lubricant  itself.  This  last  state  is  known  as  perfect 
lubrication.  The  formation  of  this  separating  film  with  in- 
creasing speed  is  probably  gradual  and  the  character  of  the 
contact  most  probably  passes  through  a  gradual  change,  from 
contact  which  is  nearly  metallic  through  successive  stages  of  par- 
•tially  fluid  contact  to  complete  fluid  separation.  The  exact 
point  at  which  perfect  lubrication  occurs  for  any  given  load, 
velocity,  and  lubricant  is  not  accurately  known,  but  what  data  are 


GENERAL   THEORY  OF   FRICTION  103 

available  will  be  given  in  connection  with  the  discussion  of  per- 
fect lubrication  which  follows.  It  is  known,  however,  that  per- 
fect lubrication  cannot  be  obtained  without  a  plentiful  supply  of 
the  lubricant,  as  in  the  case  where  a  journal  runs  in  an  oil  bath, 
or  is  supplied  by  so-called  forced  lubrication  where  the  lubricant 
is  delivered  under  pressure.  It  is  impossible  or  inconvenient, 
however,  to  lubricate  the  greater  part  of  the  rubbing  surfaces  of 
machines  in  this  manner  and,  therefore,  all  surfaces  lubricated 
by  such  means  as  simple  oil  holes,  oil  cups,  oily  pads,  etc., 
where  the  supply  of  lubricant  is  in  any  way  restricted  must  be 
considered  as  imperfectly  lubricated. 

As  already  noted,  the  exact  condition  which  will  exist  between 
such  surfaces  depends  on  the  pressure,  the  velocity  of  rubbing, 
the  supply  and  character  of  the  lubricant,  and  the  temperature  of 
the  bearing  as  affecting  the  viscosity  of  the  oil.  Naturally  where 
so  many  variables  exist,  experimental  results  are  very  discord- 
ant, and  while  an  immense  amount  of  work  has  been  done,  the 
results  only  serve  to  emphasize  the  great  variation  in  conditions 
with  change  of  these  variables.  It  is  evident,  for  instance,  that 
if  velocity  and  pressure  remain  constant,  almost  any  condition 
may  be  produced  from  metallic  contact  to  perfect  lubrication 
simply  by  varying  the  supply  of  lubricant.  The  law  of  varia- 
tion of  the  coefficient  of  friction,  with  either  varying  pressure 
or  velocity,  is  also  found  to  be  modified  by  the  rate  at  which 
oil  is  supplied.  The  generally  accepted  theories  for  imperfectly 
lubricated  bearings  running  under  average  conditions,  i.e.,  at 
normal  temperature,  and  with  good  oil  supply  from  cups  or  pads, 
are  as  follows :  * 

(a)  Starting  from  rest  with  constant  load,  the  coefficient  of 
friction  first  increases  slightly  with  increasing  velocity  and  then 
decreases,  until  at  a  velocity  somewhere  below  200  feet  per  min- 
ute (and  depending  upon  the  oil  supply)  a  minimum  value  is 
reached  (see  Fig.  i6).f  With  further  increase  of  velocity  the 

*  See  Archbutt  and  Deeley,  page  58,  and  Thurston's  "  Friction  and  Lost 
Work,"  pages  296-312. 

t  It  is  to  be  noted  that  this  discussion  and  the  coefficients  given  refer  to 
circular  bearings  and  friction  of  rotation. 


104 


MACHINE    DESIGN 


coefficient  increases  till  the  temperature  affects  the  viscosity  of 
the  lubricant  to  such  an  extent  that  abrasion  and  failure  occur. 

(b)  With  constant  velocity  and  very  light  loads  (see  Fig.  17) 
the  coefficient  of  friction  is  very  high.     As  the  load  is  increased, 
the  coefficient  decreases  very  rapidly  at  first,   and  then  more 
slowly  till  pressures  of  about  100  to  200  Ibs.  per  square  inch  are 
obtained  when  the  coefficient  again  slowly  increases. 

(c)  The  law  of  variation  of  friction  with  temperature  is  very 
complex  and  not  well  defined.     Its  general  characteristics,  how- 
ever, may  be  expressed  as  follows:   every  combination  of  pres- 
sure  and  velocity  requires  a  lubricant  of  a  certain  viscosity  for 
best  results.     At  high  speeds  and  light  loads,  a  light,  thin  oil 


500 


1400 

= 


300 


I200 


100 


000 


500 


a300 

DQ 

•d 
g200 

P-4 

.100 


..005      .01      .015      .02      .0 
Coefficient  of  Friction 

FIG.  1 6. 


.005      .01      .015      .02      .025      .08 
Coefficient  of  Friction 

FIG.  17. 


will  be  readily  drawn  in  between  the  bearings,  and  its  fluid  fric- 
tion, which  constitutes  the  greater  part  of  the  resistance  in  such 
cases,  will  be  less  than  that  of  a  heavier  oil.  Increasing  the 
temperature  of  a  lubricant  decreases  its  viscosity  and,  in  the 
above  case  therefore,  would  cause  a  decrease  in  friction.  In 
the  case  of  the  heavier  loads  and  lower  velocities,  usually  met  with 
in  machines,  an  increase  of  temperature  decreases  the  viscosity 
and  may,  owing  to  the  expulsion  of  the  lubricant,  give  an  increase 
in  friction. 

Care  should  therefore  be  used  to  obtain  an  oil  suited  to  the 
case   in   hand,   for   sometimes   a   change    of  lubricant  is  suffi- 


GENERAL   THEORY   OF  FRICTION  10$ 

cient  to  cause  great  trouble  or,  on  the  other  hand,  to  reduce  the 
temperature  of  a  bearing  that  is  heating.  The  failure  of 
imperfectly  lubricated  bearings  generally  results  from  the 
lowering  of  the  viscosity  by  increased  temperature,  so  that  the 
oil  film  is  no  longer  maintained  and  metallic  contact  and  abra- 
sion ensue. 

From  the  foregoing  it  is  evident  that  the  coefficient  of  friction 
for  imperfect  lubrication  will  necessarily  be  a  variable  quantity. 
Figs.  1 6  and  17  show  the  variation  of  /*.  for  varying  velocities 
and  pressures.  With  good  lubrication  and  moderate  velocity  it 
may  be  as  low  as  .005,  and  again  with  low  velocity  and  poor 
lubrication  it  may  rise  to  .05  or  more.  When  the  velocity  is 
exceedingly  low,  the  coefficient  approaches  that  of  static  friction 
of  lubricated  surfaces,  the  average  value  of  which  is  .15.  A  fair 
average  range  for  pressures  from  50  to  500  Ibs.,  and  velocities 
from  50  to  500  ft.  per  minute,  is  from  .02  to  .008  and,  for  pur- 
poses of  design  of  ordinary  machinery,  may  be  taken  at  .015.  It 
is  to  be  noted  that  with  imperfectly  lubricated  surfaces  and  low 
velocities  the  coefficient  of  friction  is  less  dependent  on  the 
character  of  the  lubricant,  and  more  dependent  on  the  character 
of  the  rubbing  surfaces.  The  curves  Figs.  16  and  17  are  com- 
posite curves  taken  from  a  number  of  actual  experimental  re- 
sults. They  are  not  to  be  taken  as  giving  exact  values  of  the 
coefficient  /.«,  but  serve  to  show  graphically  the  general  laws  by 
which  it  varies.  In  interpreting  such  curves  as  Fig.  17  it  must 
be  kept  in  mind  that,  while  the  coefficient  is  decreasing  or  in- 
creasing, the  actual  frictional  resistance  may  not  be  changing  in 
like  manner.  The  frictional  resistance  is  the  product  of  the 
load  and  the  coefficient  of  friction.  If,  for  instance,  the  coeffi- 
cient decreases  as  fast  as  the  load  increases,  the  frictional  resist- 
ance will  remain  constant.  The  curves  show,  however,  where 
best  results  may  be  expected  when  designing  new  machinery, 
and  throw  some  light  on  proposed  changes  in  running  speed  of 
machinery  already  installed.  They  also  indicate  the  complexity 
of  the  relation  which  exists  between  velocity,  pressure,  and  the 
coefficient  of  friction.  When  it  is  considered  that  the  tempera- 
ture also  greatly  affects  these  relations,  it  is  evident  that  a  state- 


106  MACHINE    DESIGN 

ment  of  these  relations  for  imperfect  lubrication,  in  the  form  of 
a  general  law  or  mathematical  expression,  is  impracticable,  and 
all  such  expressions  are  misleading. 

33.  Perfect  Lubrication.  It  has  been  shown  in  the  last 
article  that  any  rotating  journal  will,  by  means  of  the  molecular 
attraction  between  it  and  the  lubricant,  combined  with  the 
viscosity  of  the  lubricant,  draw  more  or  less  of  the  lubricant  in 
between  the  journal  and  bearing,  the  amount  so  drawn  in  de- 
pending on  the  velocity  and  pressure.  If  the  journal  be  allowed 
to  run  in  an  oil  bath,  or  is  otherwise  plentifully  supplied  with 
oil,  and  the  velocity  be  high  enough  for  the  pressure  carried,  it 
is  found  that  this  action  is  so  marked  that  the  rubbing  surfaces 
are  completely  separated  by  a  thin  film  of  lubricant  and  the 
friction  becomes  only  that  due  to  the  fluid  friction  of  the  lubricant 
itself. 

Mr.  Beaucamp  Tower  experimenting  with  journal  friction  (see 
Proceedings  of  Institution  of  Mechanical  Engineers,  1883)  found 
that  with  a  journal  and  bearing  arranged  as  in  Fig.  15,  the 
above  action  was  so  marked  as  to  form  a  film  of  oil  under  pres- 
sure such  that  the  load  was  completely  fluid  borne.  The  distri- 
bution of  the  pressure  in  this  film  was  found  to  be  as  indicated 
by  the  diagrams  above  the  cross-sections,  rising  to  a  maximum 
at  the  middle  and  falling  to  zero  at  the  edges  of  the  bearing. 
Mr.  Tower  succeeded  in  this  way  in  carrying  a  load  of  625 
pounds  per  square  inch  of  projected  area  at  a  velocity  of  471  ft. 
per  minute.  With  a  load  of  about  330  Ibs.  per  sq.  inch,  and  a 
velocity  of  about  150  ft.  per  minute,  a  maximum  oil  pressure  of 
625  Ibs.  was  found  near  the  middle  point  of  the  bearing.  It  has 
been  proved  mathematically,  and  verified  experimentally,  that 
the  conditions  which  exist  in  a  bearing  running  under  these 
conditions  are  as  follows:  the  journal,  being  slightly  smaller 
than  the  bore  of  the  bearing,  tends  to  be  crowded  back  from  the 
side  where  the  lubricant  is  carried  in,  as  shown  in  an  exag- 
gerated manner  in  the  figure,  giving  a  wedging  effect.  The 
pressure  is  consequently  greatest  at  a  point  a  little  more  than 
half  way  beyond  the  centre  of  loading  where  the  distance  be- 
tween surfaces  is  least. 


GENERAL    THEORY   OF   FRICTION 


The  exact  relation  which  must  exist  between  velocity  and 
pressure,  to  allow  this  pressure  film  to  form,  is  not  known  nor 
is  it  likely  that  exact  limits  can  ever  be  set.  Enough  is  known, 
however,  to  serve  as  a  general  guide  for  average  conditions. 

Professor  H.  F.  Moore  found  that  for  circular  journals  the 
minimum  limiting  values  of  pressure  and  velocity  where  the  film 
will  just  form  may  be  approximately  expressed  by  the  expression 
w=  7.47  Vi;,*  where  w  is  in  pounds  per  square  inch,  and  vm  feet 
per  minute.  The  values  given  by  this  expression  are  plotted  in 
Fig.  18,  Curve  No.  i,  and  seem  to  check  fairly  well  with  con- 
siderable other  data.  Curve  (2),  Fig.  18,  shows  the  simultaneous 
values  obtained  by  Tower  with  olive  oil,  where  frictional  resist- 
ance was  a  minimum,  indicating  that  the  film  was  at  least  well 


ouu 

JNo.2_ 

§200 

.,*** 

-- 



^ 

^—  • 
—  •      - 

^          — 

— 

No.  3 



s 

tk 

,<7 

^-^ 

^ 

47V^T 

.  

_yo.l_ 

- 

_, 

|ioo 

o 
0 

^ 

^  

^-  ' 

^^  —  — 

•           "" 

50           100          150          200          250           300          350         .400          450        50( 

.Feet  perlVTlnute 

FIG.  18. 

formed.  Curve  (3)  shows  similar  values  for  mineral  oil.  The 
values  obtained  by  Moore  are  on  the  safe  side  judged  by  Tower's 
work,  which  is  accepted  as  accurate,  and  probably  do  not  indicate 
the  very  lowest  point  at  which  a  film  will  form.  Tower  found 
that  a  film  would  form  considerably  below  the  values  given  in 
Curve  (2).  In  Moore's  experiments,  as  in  Tower's,  the  tempera- 
ture was  constant  at  90°.  Moore's  experiments  were  on  mineral 
oils.  The  results  of  Tower's  experiments  are  very  concordant 
and  conclusive,  and  show  that  the  laws  of  friction  for  perfectly 
lubricated  surfaces,  for  ordinary  speeds  and  pressures,  are  quite 
definite,  the  coefficient  of  friction  varying  as  the  square  root 
of  the  velocity  and  inversely  as  the  pressure,  very  nearly. 

*  See  American  Machinist,  Sept.  16,  1903. 


io8 


MACHINE    DESIGN 


Thus  for  olive  oil  the  relation  is  expressed    very  closely    by 

\/ '  v 

It  follows  from  this,  that  for  any  fixed  velocity  and 


=  .2 


w 


temperature  the  product  of  n  and  w  will  be  a  constant.  That  is, 
the  frictional  resistance  is  practically  constant  with  change  of 
load,  for  any  velocity.  This  was  actually  found  to  be  the  case  in 
the  experiments,  a  variation  of  pressure  per  square  inch  from  100 
to  500  not  appreciably  affecting  the  resistance.  Table  VIII  will 
serve  to  show  the  remarkable  regularity  of  the  results,  and  the  low 
values  of  the  coefficient  as  compared  with  imperfectly  lubricated 
surfaces.  Much  lower  values  have  since  been  attained  in  oil- 
testing  machines,  under  more  ideal  conditions,  but  such  low 
values  must  not  be  considered  as  attainable  under  ordinary  prac- 
tical working  conditions,  while  there  is  no  good  reason  why  such 
coefficients  as  given  below  cannot  be  obtained  in  well-constructed 
machinery. 

TABLE  VIII 

BATH   OF    RAPESEED   OIL 


COEFFICIENTS  OF  FRICTION  FOR  SPEEDS  AS  BELOW. 

Load  in 

Ibs. 
per  sq. 
Inch. 

105  ft. 
per  Min. 

'57  ft. 
per  Mm. 

209  ft. 
per  Min. 

262  ft. 
per  Min. 

314  ft. 
per  Min. 

366  ft. 

per  Min. 

419  ft- 

per  Mm. 

47i  ft. 
per  Min. 

C72 

OOIO2 

00108 

ooi  1  8 

ooi  26 

OOI  T.2 

OOI  30 

s2O 

oooo^ 

OOIO? 

OOI  I  5 

OOI  2<C 

OOI  1  3 

00142 

00148 

4.1^ 

OOOQ3 

00107 

.001  19 

OOI  T.O 

OOI4O 

OOI4.O 

ooi  58 

t 
363 



.00084 

.00960 

.OOI  10 

.00122 

.00134 

.00147 

•00155 

258 

.00107 

.00139 

.00162 

.00178 

.00195 

.00213 

.00227 

.00243 

153 

.00162 

.00200 

.00239 

.00267 

.OO3OO 

•00334 

.00367 

.00396  . 

TOO 

.00277 

•00357 

.00423 

.00503 

.00576 

.00619 

.00663 

.00714 

Tower's  experiments  have  been  amply  verified  and  may  be 
accepted  as  reliable  for  the  range  which  they  cover.  The  ex- 
periments of  Stribeck  and  Lasche  (see  Chap.  X)  have  extended 
the  range  of  knowledge  on  this  point  to  velocities  over  2,000  ft. 
per  minute.  Their  experiments  show  that  for  velocities  between 
500  and  2,000  ft.  per  minute  the  coefficient  of  friction,  for  a  given 
load  and  temperature,  varies  as  the  5th  root  of  the  velocity;  and 
beyond  2,000  ft.  is  independent  of  the  velocity.  This  point  is 


GENERAL    THEORY    OF    FRICTION  109 

discussed  still  further  in  Chap.  X  in  connection  with  the  design 
of  bearings,  where  its  principal  application  is  found. 

34.  Summary.  From  the  foregoing  discussion  the  following 
statements  may  be  made: 

(a)  The  friction  of  imperfectly  lubricated  surfaces  depends 
partly  on  the  character  of  the  surfaces  themselves,  and  in  a  greater 
degree  on  the  character  and  amount  of  the  lubricant  supplied. 

(b)  The  load  that  can  be  successfully  carried  on  an  imperfectly 
lubricated  surface  will  vary  greatly  with  the  amount  of  lubricant 
supplied,  and  must  be  kept  very  low  where  this  supply  is  re- 
stricted. 

(c)  The  friction  of  perfectly  lubricated  surfaces  depends  very 
little  on  the  character  of  the  rubbing  surfaces,  but  depends 
mainly  on  the  character  of  the  lubricant. 

(d)  The  frictional  resistance  of  perfectly  lubricated  surfaces 
is,  within  the  ordinary  limits,  independent  of  the  intensity  of 
pressure  and  dependent  only  on  the  velocity. 

(e)  The  coefficient  of  friction  of  perfectly  lubricated  surfaces, 
for  any  given  pressure  and  temperature,  varies  very  nearly  as  the 
square  root  of  the  velocity  for  velocities  up  to  500  ft.  per  minute; 
approximately  as  the  fifth  root  of  the  velocity,  for  velocities  be- 
tween 500  and  2,000  ft.  per  minute;  and  is  practically  independent 
of  the  velocity  for  values  above  2,000  ft.  per  minute. 

35-  Efficiency.  It  has  been  pointed  out  that  all  the  energy 
supplied  to  a  machine  is  not  transformed  into  useful  work,  but 
that  some  of  it  is  always  lost  in  overcoming  frictional  resistances 
and  doing  useless  work.  There  are  many  ways  in  which  energy 
losses  may  occur  in  machines,  and  a  careful  distinction  must  be 
made  between  certain  of  these  ways  in  order  to  get  a  clear 
definition  of  the  term  efficiency.  Thus  the  steam  engine  re- 
ceives its  supply  of  heat  in  the  form  of  steam  under  pressure. 
A  considerable  portion  of  the  heat  so  received  is  lost  by  con- 
densation of  steam  on  the  cooler  cylinder  walls,  and  some  escapes 
by  radiation  without  doing  any  work  whatever  on  the  piston. 
Of  the  energy  actually  applied  to  the  piston,  part  is  transformed 
into  useful  work  at  the  driving  belt,  and  part  is  lost  in  over- 


110  MACHINE    DESIGN 

coming  the  frictional  resistances  just  discussed  at  the  various 
constraining  surfaces. 

The  gas  engine  is  subject  to  similar  losses;  a  large  part  of  the 
heat  of  combustion  escaping  to  the  jacket  water  or  to  the  at- 
mosphere by  radiation,  and  doing  no  work  on  the  piston;  while 
only  a  part  of  the  energy  actually  applied  to  the  piston  reappears 
as  useful  work.  Hydraulic  and  electric  machinery  have  similar 
elements  of  loss.  The  first  class  of  these  energy  losses  might  be 
called  leakage  losses,  as  they  are  of  the  same  character  as  losses 
by  actual  leakage  of  the  medium  which  is  used  to  transmit  the 
energy.  The  losses  in  the  machine  itself  are  known  as  frictional 
losses  and  are  common  to  all  machines;  and  no  machine  can 
transform  all  the  energy  supplied  into  useful  work,  but  must  lose 
some  of  it  in  friction  or  other  wasteful  resistances. 

Efficiency  has  been  defined  (Art.  2)  as  the  ratio  of  useful 
work  to  energy  supplied;  and  from  the  above  it  appears  that  a 
machine  may  have  two  efficiencies  depending  on  whether  refer- 
ence is  had  to  total  energy  supplied,  or  to  that  portion  only  of 
the  total  energy  which  the  machine  transforms  into  useful  and 
useless  work.  These  efficiencies  are  respectively  known  as  the 
Absolute  Efficiency  and  the  Mechanical  Efficiency.  Thus, 
if  a  gas  engine  is  supplied  with  1,000  thermal  units,  and  trans- 
forms 200  units  into  useful  work,  and  50  units  into  the  useless 

work  of  friction,  its  absolute  efficiency  is  -    —  =  .20,  and  the 

1,000 

mechanical  efficiency  is  -       =  .80.     The  consideration  of  abso- 

250 

lute  efficiency  is  beyond  the  scope  of  this  work;  for  the  design 
of  many  machines  it  does  not  need  to  be  considered;  but  the 
mechanical  efficiency  can  seldom  be  neglected,  since,  in  general, 
the  amount  of  work  to  be  done  is  fixed,  and  the  source  of  energy 
must  supply  enough  more  energy  than  this  to  compensate  for  the 
frictional  losses  of  the  machine. 

The  mechanical  efficiency  of  any  train  of  mechanism  is  the 
continued  product  of  the  efficiencies*  of  all  the  several  pairs  of 

*  It  may  be  noted  in  passing  that  the  term  efficiency  is  used  in  a  number  of  ways 
other  than  as  the  ratio  of  work  done  to  energy  expended.    Thus  the  strength  of  a 


GENERAL  THEORY  OF   FRICTION  III 

constraining  surfaces  in  the  train  at  which  frictional  losses 
occur.  Let  any  machine  have  n  pairs  of  such  surfaces,  and  let 
their  respective  efficiencies  be  e,  elt  e2,  e3,  e4,  -  —  eu.  Let  E 
be  the  mechanical  efficiency  of  the  whole  machine,  and  let  K  be 
the  total  amount  of  energy  available  for  transformation  into 
either  useful  or  useless  work.  Then,  the  amount  of  energy  which 
the  first  pair  of  constraining  surfaces  delivers  to  the  second  is 
K  X  e,  and  the  amount  which  the  second  delivers  to  the  third  is 
Ke  X  elt  and  so  on,  till  the  amount  of  energy  delivered  by  the 

last  element  (or  the  work  done)  is  K  (e  X  el  X  e2 ej. 

But  the  mechanical  efficiency  of  the  train  is 

_        work  done        _  K  (e  X  et  X  e2  —  —eu) 

energy    supplied  K 

-(eXe.Xe,-         -ej. 

A  machine  may  consist  of  several  trains  of  mechanism.  If 
these  several  trains  are  arranged  in  series  so  that  the  energy 
passes  from  one  to  another  consecutively,  the  efficiency  of  the 
whole  machine,  by  reasoning  similar  to  that  in  the  last  paragraph, 
is  the  continued  product  of  the  efficiencies  of  the  several  trains 
of  mechanism.  If,  however,  the  trains  are  arranged  in  parallel 
so  that  the  total  energy  is  transmitted  simultaneously  through 
several  trains  of  mechanism,  each  train  transmitting  only  a  por- 
tion of  the  energy,  the  above  reasoning  for  the  efficiency  of  the 
whole  machine  does  not  hold.  If  the  amount  of  energy  supplied 
to  each  train  is  known,  the  amount  of  work  which  it  will  deliver 
can  be  computed  as  above.  The  sum  of  all  the  work,  delivered 
by  all  the  trains,  divided  by  the  total  energy  supplied,  will  be  the 
efficiency  of  the  whole  machine. 

If,  therefore,  the  efficiencies  of  the  several  constraining  sur- 
faces of  a  machine  are  known,  the  mechanical  efficiency  of  the 

riveted  joint,  compared  to  the  strength  of  the  original  unpunched  plate,  is  called 
the  efficiency  of  the  joint,  when  what  really  is  meant  is  its  relative  strength.  Again, 
in  an  air  compressor,  the  ratio  of  the  air  actually  discharged  per  stroke,  to  the  whole 
amount  raised  to  the  required  pressure  per  stroke,  is  called  the  volumetric  efficiency. 
It  is  evident  that  such  efficiencies  are  of  a  different  character  from  those  discussed 
above  and  do  not  enter  into  the  calculations  of  the  efficiency  of  the  machine,  as  a 
whole,  in  the  manner  indicated  above. 


112  MACHINE    DESIGN 

whole  machine  can  be  calculated.  The  mechanical  efficiency  of 
any  machine  element  is,  however,  a  variable  quantity;  for  the 
coefficient  of  friction  of  any  pair  of  constraining  surfaces  will 
vary  with  the  lubricant  and  its  method  of  application,  the  tem- 
perature, the  alignment  of  the  surfaces,  the  velocity  of  rubbing, 
and  the  bearing  pressure.  Furthermore,  when  all  other  condi- 
tions are  constant,  the  same  pair  of  constraining  surfaces  will 
have  an  entirely  different  efficiency  for  the  same  amount  of 
power  transmitted,  depending  on  the  manner  in  which  the  load 
is  applied.  Thus,  consider  a  simple  wheel  and  axle  driven  by  a 
belt  on  the  periphery  of  the  wheel.  With  a  given  diameter  of 
wheel,  the  transmission  of  a  given  amount  of  power  will  bring  a 
certain  definite  frictional  load  on  the  bearings.  If,  however, 
the  diameter  of  the  wheel  is  doubled,  the  belt  speed  is  increased 
in  a  like  ratio,  and  the  belt  tension  will,  for  the  same  power 
transmitted,  be  one-half  of  the  former  value;  and,  as  a  conse- 
quence, the  frictional  resistance  at  the  bearings  will  be  reduced 
to  one-half  the  original  value,  the  revolutions  remaining  con- 
stant. 

In  general,  therefore,  it  is  impossible  to  calculate  precisely 
from  the  analysis  of  a  design  what  the  mechanical  efficiency 
will  be,  particularly  if  the  mechanism  is  at  all  complicated, 
though  a  reasonable  approximation  is  possible.  If  machines  of 
a  similar  type  have  been  built,  it  is  far  more  accurate  to  base 
the  design  of  new  ones  on  efficiency  tests  made  on  those  already 
in  existence.  For  all  standard  machines  such  tests  have  been 
made,  and  the  recorded  results  form  a  valuable  basis  for  the  de- 
sign of  new  machines  of  like  characteristics.  But  when  a  ma- 
chine of  a  new  type  is  to  be  designed,  and  no  recorded  tests  are 
to  be  had  that  will  give  any  information  as  to  the  probable  ef- 
ficiency, an  estimate  must  often  be  made  and  the  efficiency  cal- 
culated as  outlined  above.  In  general,  a  close  approximation 
can  be  made,  and  the  making  of  such  estimates  is  a  great  aid  to 
the  development  of  that  judgment  in  such  matters,  which  comes 
only  with  experience.  In  such  cases  a  knowledge  of  the  ef- 
ficiencies of  various  machine  elements  becomes  necessary.  If 
the  coefficient  of  friction  for  any  constraining  surface  could  be 


GENERAL   THEORY   OF   FRICTION  113 

accurately  determined,  it  would  be  possible  to  calculate  its 
efficiency  with  some  degree  of  certainty.  But,  as  before  noted, 
the  quantity  varies  with  the  velocity  of  rubbing,  with  changes  in 
bearing  pressures,  etc.,  and  such  methods  of  computation  are 
necessarily  cumbersome  and  to  be  attempted  only  where  a  very 
close  estimate  is  required. 

The  following  are  rough  average  values  of  the  efficiencies  of 
the  most  common  elements.  For  more  accurate  values  the  stu- 
dent is  referred  to  the  respective  discussions  of  these  various 
elements  which  follow: 

Common  Bearing,  singly 96-98 

Common  Bearing,  long  lines  of  shafting 95 

Roller  Bearing 98 

Ball  Bearings 99 

Spur  Gear  Cast  Teeth,  including  bearings 93 

Spur  Gear  Cut  Teeth,  including  bearings 96 

Bevel  Gear  Cast  Teeth,  including  bearings 92 

Bevel  Gear  Cut  Teeth,  including  bearings 95 

Worm  Gear,  varies  with  thread  angle,  see  Art.  54 

Belting 96-98 

Pin-connected  Chains,  as  used  on  bicycles 95~97 

High  Grade  Transmission  Chains 97~99 


CHAPTER  V 
SPRINGS 

36.  Distinguishing    Characteristic    of   Springs.     Springs    are 
characterized  by  a  considerable  distortion  under  a  moderate  load. 
Every  machine  member  is,  in  a  sense,  a  spring,  for  no  material 
is  absolutely  rigid  and  the  application  of  a  load  always  produces 
stress  and  accompanying  strain.     By  proper  selection  and  distri- 
bution of  material  it  is  possible  to  control  (within  wide  limits) 
the  degree  of  distortion  under  a  given  load. 

An  absolutely  rigid  material  would  be  practically  unfit  for  the 
construction  of  any  member  subject  to  other  than  a  perfectly 
quiescent  load;  for  (as  shown  in  Art.  24)  the  stress  due  to  a  sud- 
denly applied  load  would  be  infinite  if  the  corresponding  distor- 
tion of  the  member  were  zero. 

While  it  is  usually  desirable  to  restrict  the  distortions 
of  most  machine  parts  to  very  small  magnitudes,  there  are 
many  cases  in  which  considerable  distortion  under  moderate 
load  is  desirable  or  essential.  To  meet  this  last  requirement 
the  member  is  often  given  some  one  of  the  forms  commonly 
called  springs. 

37.  The  Principal  Applications  of  Springs.     Springs  are  in 
common  use: 

I.  For  weighing  forces;  as  in  spring  balances,  dynamome- 
ters, etc. 

II.  For  controlling  the  motions  of  members  of  a  mechanism 
which  would  otherwise  be  incompletely  constrained;  for  example, 
in  maintaining  contact  between  a  cam  and  its  follower.  This 
constitutes  what  Reuleaux  has  called  "  force  closure." 

III.  For  absorbing  energy  due  to  the  sudden  application  of  a 
force  (shock) ;  as  in  the  springs  of  railway  cars,  etc. 

IV.  As  a  means  of  storing  energy,  or  as  a  secondary  source  of 
energy;  as  in  clocks,  etc. 

114 


SPRINGS 

An  important  class  of  mechanisms  in  which  springs  are  used 
to  weigh  forces  is  a  common  type  of  governor  for  regulating  the 
speed  of  engines  or  other  motors.  In  those  governors  which  use 
springs  to  oppose  the  centrifugal,  or  other  inertia  actions,  the 
springs  automatically  weigh  forces  which  are  functions  of  speed, 
or  of  change  of  speed.'  The  links,  or  other  connections,  which 
move  relative  to  the  shaft  with  any  variation  of  the  above  forces, 
correspond  to  the  indicating  mechanism  of  ordinary  weighing 
devices. 

The  first  of  the  above-mentioned  applications — the  weighing  of 
forces— is  usually  the  most  exacting  as  to  the  relation  between 
the  load  and  the  distortion  of  the  spring  throughout  the  range  of 
action.  In  the  second  and  third  classes  of  application,  it  is  fre- 
quently only  required  that  the  maximum  load  and  distortion 
shall  lie  within  certain  limits,  which  often  need  not  be  very  pre- 
cisely defined.  The  use  of  springs  for  storing  energy  (as  the 
term  spring  is  ordinarily  understood)  is  almost  wholly  confined  to 
light  mechanisms  or  pieces  of  apparatus  requiring  but  little 
power  to  operate  them. 

38.  Materials  of  Springs.  Springs  are  usually  of  metal; 
although  other  solid  substances,  as  wood,  are  sometimes  used. 
A  high  grade  of  steel,  designated  as  spring  steel,  is  the-  most 
common  material  for  heavy  springs,  but  brass  (or  some  other 
alloy)  is  often  used  for  lighter  ones. 

A  confined  quantity  of  air,  or  other  compressible  fluid,  is  used 
in  many  important  applications  to  perform  the  office  of  a  spring. 
The  air-chamber  of  a  pump  with  its  inclosed  air  is  a  familiar  ex- 
ample of  what  may  be  called  a  fluid  spring  used  to  reduce  shock 
("water  hammer").  The  characteristic  distortion  of  the  solid 
springs  is  a  change  in  form  rather  than  of  volume;  while  the 
fluid  springs  are  characterized  by  a  change  of  volume  with  inci- 
dental change  of  form. 

Soft-rubber  cushions,  or  buffers,  are  not  infrequently  employed 
as  springs,  and  these  are  in  some  respects  intermediate  in  their 
action  to  the  two  classes  mentioned  above.  It  is  usually  not  nec- 
essary, in  these  simple  buffers,  or  cushions,  to  secure  a  very  exact 
relation  between  the  loads  and  the  distortions  under  such  loads. 


n6 


MACHINE    DESIGN 


The  discussion  of  the  confined  gases  (fluid  springs)  is  not  within 
the  scope  of  the  present  work;  hence  the  following  treatment 
will  be  limited  to  solid  springs. 

39.  Forms  of  Solid  Springs.     Springs  may  be  subjected  to 
actions  which  extend,  shorten,  twist,  or  bend  them,  producing 


FIG.  29. 


stresses  in  the  material,  the  character  of  which  depends  both  upon 
the  form  of  the  spring  and  upon  the  manner  of  applying  the  load. 
I.  Flat  Springs  are  essentially  beams,  either  cantilevers,  or 
with  more  than  one  support.  These  springs  are  subjected  to 
flexure  when  the  load  is  applied,  and  the  resultant  stresses  are 
tension  in  certain  portions  of  the  material,  and  compression  in 
others,  with  a  transverse  shear,  as  in  all  beams;  the  shear  may 


SPRINGS 


117 


usually  be  neglected  in  computations.  The  ordinary  beam 
formulae  for  strength  and  rigidity  may  be  applied  to  flat  springs, 
with  constants  appropriate  to  the  particular  material  and  form  of 
beam  used. 

Flat  springs  may  be  simple  prismatic  strips,  of  uniform  cross- 
section  (Fig.  19  or  22),  although  it  is  preferable  that  the  form 
of  such  springs  approximate  those  of  the  " uniform  strength" 
beams  (Figs.  20  or  21;  23  or  24). 

It  is  often  desirable  or  practically  necessary  to  build  up  these 
springs  of  several  layers,  leaves,  or  plates,  producing  a  laminated 
spring.  It  will  appear  from  the  discussion  of  these  laminated 


FIG.  30.  FIG.  30  (a). 

springs  that  they  may  be  properly  treated  as  a  modification  of  one 
form  of  "uniform  strength"  beam.  The  neutral  surface  of  the 
beam  used  as  a  spring  may  be  initially  curved,  either  to  clear 
other  bodies,  or  to  give  the  spring  an  advantageous  form  when  it 
is  under  normal  load.  See  Fig.  27. 

Two  or  more  springs  may  be  compounded,  as  in  the  "  ellipti- 
cal" springs  or  in  the  platform  springs  frequently  used  under 
carriages.  In  such  cases,  each  spring  may  be  computed  sepa- 
rately, and  the  total  deflection  is  the  sum  of  the  deflections  of  the 
separate  springs  of  the  set. 

II.  Helical,  or  Coil  Springs  are  most  commonly  used  to  resist 
actions  which  extend,  shorten,  or  twist  the  spring  relatively  to 
its  longitudinal  axis.  These  are  sometimes  improperly  called 
spiral  springs. 


Il8  MACHINE    DESIGN 

The  stress  in  the  wire  (or  rod)  of  which  a  helical  spring  is 
made  is  somewhat  complex,  consisting  of  torsion  combined  with 
tension  or  compression,  or  both.  In  a  "pull  spring,"  one  which 
is  extended  longitudinally  under  the  load,  the  predominating 
stress  (with  ordinary  proportions)  is  a  torsion,  and  there  is  a 
secondary  tensile  stress  in  the  wire.  In  a  "push  spring,"  one 
which  is  shortened  by  the  load,  the  predominating  stress  is  tor- 
sion, with  a  secondary  compressive  stress.  When  the  helical 
spring  is  subjected  to  an  action  which  twists  the  spring  (as  a 
whole)  the  principal  stress  in  the  wire  is  that  due  to  flexure 
(tension  and  compression  in  opposite  fibres)  and  the  secondary 
stress  is  torsion. 

Helical  springs  are  sometimes  arranged  in  "nests,"  springs  of 
smaller  diameter  being  placed  within  those  of  larger  diameter, 
(Fig.  30).  In  these  cases,  the  different  springs  of  a  set  are  com- 
puted separately.  This  last  arrangement  is  common  practice  in 
car  trucks. 

III.  Spiral  Springs,  properly  so  called  are  those  of  the  form  of 
the  familiar  clock  spring.     These  are  best  adapted  for  a  twist 
relative  to  the  axis  of  the  spiral,  and  are  usually  employed  when 
a  very  large  angle  of  torsion  between  the  two  connections  is 
necessary.     In  this  form  of  spring,  the  stress  in  the  material  is 
that  due  to  flexure :  or  tensile  and  compressive  stress  on  opposite 
sides  of  the  neutral  axis. 

IV.  Helico-Spiral  Springs.     The  form  of  spring  represented 
by  the  common  upholstery  spring  may  be  looked  upon  as  a  spiral 
spring  which  has  been  elongated,  and  given  a  permanent  set,  in 
the  direction  of  its  axis;   or  it  may  be  considered  as  a  modified 
helical  spring  in  which  the  radii  of  the  successive  coils  are  not 
equal.     It  is  thus  intermediate  between  the  two  preceding  classes. 
This  last  form  is  not  usual  in  machine  construction;   though  it 
has  the   advantage  over   the  common   helical   spring  of  con- 
siderable   lateral    resistance,    and    it    may    be    employed     to 
advantage  where   it    is    difficult    or    undesirable    otherwise    to 
constrain  the  spring  against  buckling.     This  spring  is  used  only 
as  a  push  spring,  to  resist  a  compressive  action.     The  springs 
used  on  the  ordinary  disc  valves  of  pumps  are  often  of  this 


SPRINGS 

form,  as  they  will  close  up  flat  between  the  valve  and  guard. 
Car  springs  are  sometimes  made  of  a  flat  strip  or  ribbon  of  steel 
wound  in  this  general  form,  with  the  edges  of  the  strip  parallel 
to  the  axis  of  the  spring. 

V.  Occasionally  straight  rods,  usually  of  circular  or  rectangular 
cross-sections,  are  employed  to  resist  torsion  relative  to  their 
longitudinal  axis.  These  are  comparatively  stiff  springs,  and 
the  stress  is,  of  course,  torsional.  Every  line  of  shafting  is 
necessarily  a  spring,  in  this  sense. 

The  following  summary  gives  the  ordinary  forms  of  solid 
springs;  the  kinds  of  loading  to  which  they  are  subjected;  and 
the  predominating  stresses  resulting  from  the  different  loads. 


GENERAL    SUMMARY   OF    SPRINGS 


Form  of  Spring. 

Load  Action. 

Predominating  Stress. 

Flat  Spring. 
Helical  Spring. 

u                « 

Spiral         " 

Flexure  or  Bending. 
Extension,  Pull. 
Compression,  Push. 
Torsion,  Twist. 
Torsion,  Twist. 

Tension  and  Compression. 
Torsion  (plus). 
Torsion  (minus). 
Tension  and  Compression. 
Tension  and  Compression. 

41.  Computations  of  Simple  Flat  Springs.  The  following 
notation  will  be  used  in  treating  of  flat  springs  with  rectangular 
cross-sections. 

P  =  load  applied  to  the  spring. 

/  =  free  length  of  the  spring. 
p  =  intensity  of  stress  in  outer  fibres. 

/  =  moment  of  inertia  of  most  strained  section. 

h  =  dimension  of  this  section  in  plane  of  flexure. 

b  =  dimension  of  this  section  perpendicular  to  plane  of  flexure. 
£  =  modulus  of  elasticity  of  material. 

d  =  deflection  of  the  spring. 

The  six  forms  of  rectangular  section  beams,  shown  by  Figs,  19 
to  24,  are  the  most  important  of  those  used  as  simple  flat  springs. 
These  will  be  designated  Type  I,  II,  etc.,  as  in  the  following 


I2O 


MACHINE    DESIGN 


table,  which  gives  the  constants  to  be  substituted  in  the  general 
formulae  for  computations  relating  to  each  type. 


TABLE   IX 


TYPE. 

COEFFICIENTS. 

A 

) 

B 

K 

C 

I 

As  per  Fig.  19 

1 

A 

1 

i 

i 

II 

"      "      "      20 

2 
~5 

A 

1 

i 

1 

III 

"         "         "         21 

,    t 

£ 

i 

\ 

1 

IV 

"        "         "        22 

i 

i 

4 

| 

6 

V 

-    "         "'         "         23 

I 

i 

6 

i 

6 

VI 

»         "         "        24 

\ 

f 

8 

* 

6 

The  theory  of  strength  against  flexure  (equation  J  and  tables 
i  and  2)  gives:  For  rectangular  section  beams  supported  at  the 
ends  and  loaded  at  the  middle  (Types  I,  II,  III). 

-Pl  =  ^-pbh*.'.Pl  =  -pbh2     ...      .     (i) 
40  3 

For  the  rectangular  section  cantilevers,  with  load  at  free  end, 

Pl^'-pbh*    .      ,      .      .      .      .      (2) 

Or  the  general  formula  for  the  strength  of  rectangular  section 
beams  may  be  written 

Pl-*P*V.    ......      (3) 

In  which  the  coefficient  A  has  the  values  given  in  the  Table. 
The  theory  of  elasticity  of  beams  gives 


•  El  ' 


or  for  rectangular  cross-sections 

3  =  B 


PI3 


(4) 


(5) 


Ebh3 

In  which  p  and  B  are  as  given  in  the  Table,  for  the  types  under 
consideration. 

The  last  equation  (5)  may  be  used  for  all  computations  as  to 
rigidity  of  flat  springs  (beams),  provided  the  elastic  limit  is  not 


SPRINGS  121 

exceeded.  The  only  constant  for  the  material  which  enters  this 
expression  is  the  modulus  of  elasticity  (£);  this  is  simply  the 
ratio  of  stress  to  strain  which  holds  up  to,  but  not  beyond,  the 
elastic  limit;  hence  any  computation  made  by  this  formula 
should  be  checked  for  safety.  Equation  (3)  may  be  used  for  this 
purpose.  To  illustrate,  assume  that  a  rectangular  section  pris- 
matic spring  (Type  I)  has  a  length  between  supports  of  /  =  30"; 
the  load  at  the  middle  is  P  =  i,ooo  Ibs.;  the  deflection  under  this 
load  is  to  be  ^  =  1.5  inches;  and  the  spring  is  made  of  a  single 
strip  of  steel  H  inch  thick  (h).  Required  the  breadth  (b)  of  the 
spring,  assuming  the  modulus  of  elasticity,  £  =  30,000,000. 
From  eq.  (5) : — 

PI3        i        1,000  X  27,000  X  S12 

b  =  B  -=—7-3  =  -  X  -  — ^—  =  2.84  +  inches. 

Edh3       4       30,000,000  X  1.5  X  27 

This  gives  the  width  of  spring  for  the  required  relation  of  the 
deflection  to  load;  that  is,  it  gives  a  spring  of  the  required  stiff- 
ness, provided  the  stress  produced  does  not  exceed  the  elastic 
limit.  It  is  necessary  to  check  the  spring  as  found  above,  for  if 
the  elastic  stress  is  passed,  the  spring  not  only  takes  a  permanent 
set,  but  the  required  ratio  of  the  load  to  the  deflection  will  not  be 
secured.  On  the  other  hand,  it  is  often  important  for  economy 
of  material  to  use  as  light  a  spring  as  is  consistent  with  safety; 
or,  in  other  words,  it  is  important  not  to  have  too  low  a  working 
stress  under  the  maximum  load. 

From  eq.  (3)  :- 

PI         3  X  1000  X  30  X  64 

p  =  =  x — £ =  T  12x00  Ibs.  per  sq.  inch. 

Abh2  2  x  2.84  X  9 

This  stress  is  beyond  the  elastic  limit  of  any  ordinary  grade  of 
steel,  hence  it  is  probable  that  some  different  form  of  spring  should 
be  used.  A  change  could  be  assumed,  as  in  the  thickness  of  the 
plate,  and  new  computations  made  with  the  new  data.  A  thinner 
plate  would  reduce  the  stress,  but  it  would  demand  a  wider  spring 
for  the  required  stiffness.  A  more  general  method  will  now  be 
given,  by  which  it  is  possible  to  determine  the  proper  spring  for 
given  requirements  without  the  necessity  of  successive  trial  com- 
putations. 


122  MACHINE    DESIGN 

From  eq.  (3) :— 

b»_lLL...b».L»    ....    (6) 

From  eq.  (5) :— 

»'-*£  ....:.   <„ 

From  eqs.  (6)  and  (7)  :— 

.  Plh  _BPl\ 
A  p          Ed   ' 

.'.  h 

From  eq.  (3)  :- 

i    PI  PI 

~~~AjW~    '~ptf 

The  two  equations  (8)  and  (9)  are  in  convenient  form  for  de- 
signing a  flat  spring  when  the  span  (/),  deflection  (<5),  load  (P), 
and  the  material  are  given.  Example :  The  span  of  a  rectangular 
section  prismatic  flat  spring  (Type  I)  is  30  inches;  and  a  load  of 
1,000  Ibs.  applied  at  the  middle  is  to  cause  a  deflection  of  1.5 
inches. 

If  the  modulus  of  elasticity  be  30,000,000  and  the  safe  maxi- 
mum working  stress  be  taken  at  50,000  Ibs.  per  sq»  in.,*  required 
the  dimensions  of  the  cross-section,  h  and  b. 
From  eq.  (8)  :— 

„  p?        i  50,000  X  900  i  . 

7,  If     I-  _    v/  J      7 " ir»/^k 

iv    —    /V      _  r  -    ~       /\  1 11L II. 

E  '>       6       30,000,000  X  1.5       6 

Taking  h  =  ^  inch,  to  use  a  regular  size  of  stock,  p  will  be 
somewhat  less  than  50,000,  or 

p  :  50,000  ;: -fy  : ~;  .*./>  =  47>ooo. 

From  eq.  (9) : — 

^  P  I       3       1000  X  30  X  1024 
b  =  C  —T-.  =  --  X  -  -  =  39.2  inches. 

p  h2       2  47,000  X  25 

*  If  the  spring  is  provided  with  stops  to  prevent  deflection  beyond  a  certain 
amount,  the  stress  due  to  such  deflection  may  be  nearly  equal  to  the  elastic  limit  of 
the  material.  A  very  small  factor  of  safety  is  all  that  is  necessary. 


SPRINGS  123 

If  this  width  is  inadmissible,  a  laminated  or  plate  spring  may 
be  used.  See  next  article. 

It  will  be  noted  that  equation  (8)  does  not  directly  involve 
either  the  load  P  or  the  breadth  of  spring  b.  It  is  evident  that  if 
a  beam  (flat  spring)  of  given  span  (/),  and  thickness  (h),  is 
caused  to  deflect  a  given  amount  (<S),  the  outer  fibres  will  undergo 
a  definite  strain  which  is  not  dependent  upon  the  width  of  the 
beam  (£),  nor  upon  the  force  required  to  produce  this  change  in 
relative  positions  of  the  molecules.  As  the  unit  strain  multiplied 
by  the  modulus  of  elasticity  equals  the  unit  stress,  it  follows  that 
this  stress  may  be  computed  from  /,  h,  and  S  (which  determine 
the  strain),  in  connection  with  E.  If  the  breadth  of  the  beam 
(b)  is  increased,  the  force  (P)  required  to  produce  the  given  de- 
flection (8)  will  be  proportionately  increased,  but  the  intensity  of 
stress  is  not  affected  by  these  changes  alone. 

This  same  conclusion  may  be  reached  from  the  following  rela- 
tion,* in  which  />  =  the  radius  of  curvature  due  to  load. 

EL    EI  .  iL  SUL  (IO) 

"~M  r  y2  h  ~  2p 

Eh 
.'.  p  =  - (u) 

2  /> 

It  appears  from  eq.  (n)  that  the  stress  is  simply  proportional 
to  the  thickness  (h)  and  the  radius  of  curvature  (/>),  for  any 
given  value  of  E.  The  span  /,  and  the  deflection  <*,  determine  p, 
so  that  eq.  (10)  or  (n)  may  take  the  place  of  eq.  (8).  Equations 
(10)  and  (n)  are  important  in  connection  with  the  theory  of 
laminated  springs. 

42.  Laminated,  or  Plate,  Springs.  It  was  shown  in  the  preced- 
ing article  that  the  maximum  thickness  of  a  simple  flat  spring  is 
fixed  when  the  span,  deflection,  and  modulus  of  elasticity  are 
known,  and  the  intensity  or  working  stress  has  been  assigned. 
[See  eq.(8).]  With  the  value  of  the  thickness  (h)  thus  limited 
it  will  frequently  happen  that  a  simple  spring  will  require  ex- 
cessive breadth  (b)  to  sustain  the  given  load,  and  it  is  often 
necessary  to  use  a  spring  built  up  of  several  plates  or  leaves. 

*  See  Church's  "Mechanics,"  page  250. 


124  MACHINE    DESIGN 

Example:  P  =  1,000  Ibs.;  /  =  3o";  p  =  60,000  Ibs.  per  sq. 
in.;  8  =  2"j  and  £  =  30,000,000.  A  simple  prismatic  spring  of 
rectangular  section,  with  load  at  the  middle  of  the  span  (Type  I)  , 
to  meet  the  above  requirements  would  have: 

p  I2        i         60,000  X  900 
h  =  K-=^-  =  -  X  —  -  =  .1^  inch. 

E  $        6       30,000,000  x  2 

,,  P  I       i  1,000  X  30 

b  =  C—r2  =  ±Xr-  -  =  33^  inches. 

p  h2       2       60,000  X  .0225 

This  spring,  consisting  of  a  plate  .15  inch  thick  and  33  % 
inches  wide,  with  a  span  of  30  inches,  is  evidently  an  impractica- 
ble one  for  any  ordinary  case.  Suppose  this  plate  be  split  into  six 
strips  of  equal  width,  each  33.  3  -v-6  =  5.5"  wide,  and  that  these 
strips  are  piled  upon  each  other  as  in  Fig.  25;  then,  except  for 
friction  between  the  various  strips,  the  spring  would  be  exactly 
equivalent?  as  to  stiffness  and  intensity  of  stress,  to  the  simple 
spring  computed  above.  While  the  form  of  laminated  spring 
which  has  just  been  developed  might  answer  in  some  cases,  an- 
other form,  based  upon  the  "uniform  strength"  beam  (Type  II), 
is  much  better  for  the  ordinary  conditions.  It  may  be  developed 
as  follows,  taking  the  same  data  as  the  preceding  example  except 
that  the  spring  is  to  be  of  Type  II,  Fig.  20. 

In  the  simple  spring,  Type  II,  Table  IX 


p?        I          60,000  X  900 

inches. 

Ed        4  ^  30,000,000  X  2 

Pl  _3  v      i-°°oX30 

inrhps 

A  laminated  spring  for  the  case  under  consideration  may  be 
derived  from  this  simple  spring  by  imagining  the  lozenge-shaped 
plate  to  be  cut  into  strips  which  are  piled  one  upon  another  as  in- 
dicated in  Fig.  26.  The  thickness  of  .225  inches  does  not  corre- 
spond to  a  regular  commercial  size  of  stock,  however,  and  it  will 
usually  be  better  to  modify  the  spring  to  permit  using  standard 
stock.  If  a  thickness  of  %"  be  assumed  for  the  leaves  or  plates, 
the  stress,  as  found  from  eq.  (8)  of  the  preceding  article  becomes: 


SPRINGS  125 

4  X  .2=;  X  30,000,000  X  2 

P  =     r  n    =  =  66,700. 

KP  900 

If  this  stress  is  considered  too  great,  steel  Ty  thick  might  be 

4  X  3  X  30,000,000  X  2 
used,  when  p  =  -        — — -  =  50,000. 

With  h  =  Ty,  and  p  =  50,000, 

C  30  X  256  _ 

V  >-•'  7    O  X  \  2>«O        • 

£  Jr      2  50,000  x  9 

If  this  spring,  30"  span,  Ty  thick,  and  25.6"  wide  at  the 
middle,  be  replaced  by  5  equivalent  strips,  each  25.6-7-5  =  5.11" 
wide  (nearly  $l/%"),  see  Fig.  26,  a  laminated  spring  of  good  form 
and  practical  dimensions  will  result.  In  cases  where  the  maxi- 
mum allowable  width  of  spring  is  fixed,  a  larger  number  of 
plates  may  be  necessary.  Thus,  in  the  preceding  problem,  if  the 
spring  width  must  be  kept  within  4%",  it  is  necessary  to  use  6 
plates,  each  25.6-^6  =  4.27"  wide.  In  actual  springs,  the  usual 
construction  is  that  shown  by  Fig.  27,  in  which  the  several  plates 
have  the  ends  cut  square  across  instead  of  terminating  in  tri- 
angles. These  springs  approximate  uniform  strength  beams,  and 
may  be  computed  by  equations  (8)  and  (9)  of  Art.  41,  remember- 
ing that  b  is  the  breadth  of  the  equivalent  simple  spring.  Or,  if 
n  is  the  number  of  plates  and  bl  the  breadth  of  each  plate  in  the 
laminated  spring,  n  b^b. 

The  last  of  these  formulae,  eq.  (9),  is  not  strictly  applicable 
when  the  ends  of  the  plates  are  cut  square  across;  but  it  may 
generally  be  used  with  sufficient  accuracy,  provided  the  succes- 
sive plates  are  regularly  shortened  by  uniform  amounts.  It  is 
quite  common  practice  to  have  two  or  more  of  the  plates  extend 
the  full  length  of  the  spring.  This  construction  makes  the  spring 
a  combination  of  the  triangular  and  prismatic  types  (Type  II 
and  Type  I,  or  Type  V  and  Type  IV,  depending  upon  whether 
the  spring  is  supported  at  the  ends,  or  is  a  cantilever).  Mr.  G. 
R.  Henderson  in  discussing  the  cantilever  form  (Trans.  A.  S. 


126  MACHINE    DESIGN 

M.E.,  Vol.  XVI),  says:— "For  a  spring  with  all  the  plates  full 
length  we  would  have  (see  eq.  5) 

,          4P13 
Enb.h3 

so  for  one-fourth  of  the  leaves  full  length,  the  deflection  would  be 
decreased  approximately  one-fourth  of  the  difference  between 

6P13  4P13  5.5  P/3/' 

E  n  b,h3  3       Enbh3  Cr  Enbh3 

By  similar  reasoning,  for  a  spring  loaded  at  the  middle  and 
supported  at  the  ends,  with  one-fourth  the  plates  extending  the 
whole  length  of  the  spring, 

d  _  ii      PI3 
~  32  Enb^h3' 

This  may  be  otherwise  stated  as  follows: 

When  the  number  of  full-length  leaves  is  one-fourth  the  total 
number  of  leaves  in  the  spring,  use  ^1  B  instead  of  B  and  |~i  K 
instead  of  K  in  equations  (5)  and  (8)  of  the  preceding  article; 
the  values  of  B  and  K  being  those  given  for  the  triangular  forms, 
Type  II  or  Type  V,  as  the  case  may  be. 

The  spring  shown  in  Fig.  27  is  initially  curved  (when  free), 
which  is  common  practice.  The  best  results  are  obtained  by 
having  the  plates  straight  when  the  spring  is  under  its  normal  full 
load  (if  this  is  practicable)  because  the  sliding  of  the  plates  upon 
each  other,  with  the  vibrations,  is  then  reduced  to  a  minimum. 
The  several  plates  of  a  laminated  spring  are  usually  secured  by 
a  band  shrunk  around  them  at  the  middle  of  the  span.  This  band 
stiffens  the  spring  at  the  middle,  and  one-half  the  length  of  the 
band  (%  I,  Fig.  27)  may  be  deducted  from  the  full  span  to  give 
the  effective  span  to  be  used  as  /  in  the  above  formulae.  It  is 
not  uncommon  to  make  the  longest  plate  thicker  than  the  others, 
if  but  one  plate  is  given  the  full  length  of  the  spring.  This  cannot 
be  looked  upon  as  desirable  practice,  however,  as  all  of  the  plates 
are  subjected  to  the  same  change  in  radius  of  curvature;  hence  the 
thicker  plate  is  subjected  to  the  greater  stress.  See  equation  (n). 

The  following  formulas  (derived  from  the  preceding)  may  be 
used  in  computing  flat  springs;  but  it  must  be  remembered  that 


SPRINGS  127 

there  is  always  liability  of  considerable  variation  in  the  modulus 
of  elasticity,  hence  such  computations  can  only  be  expected  to 
give  approximations  to  the  deflections  which  will  be  observed  by 
tests  of  actual  springs.  These  computations  will  be  sufficiently 
exact  for  many  purposes;  but  when  it  is  important  accurately  to 
determine  the  scale  of  the  spring  (ratio  of  deflection  to  load), 
actual  tests  must  be  made.  In  using  these  formulae  the  follow- 
ing rules  should  be  observed. 

I.  When  the  several  plates  are  secured  by  a  band  shrunk,  or 
forced,  over  them,  one-half  the  length  of  the  band  is  to  be  sub- 
tracted from  the  length  of  the  spring  to  get  the  effective  length 
of  the  spring. 

II.  When  the  plates  have  different  thicknesses,   the  stress 
should  be  computed  for  the  plate  having  the  maximum  thickness. 

III.  If  more  than  one  plate  has  the  full  length  of  the  spring, 
an  appropriate  modification  of  the  values  of  the  coefficients  B  and 
K  should  be  made.     Thus,  when  one-fourth  of  the  total  number 
of  plates  are  full  length,  \^  B  and  {^  K  should  be  used  instead  of 
B  and  K  (Type  II  or  V)  in  equations  I,  II,  III,  and  IV,  below. 

EQUATIONS. 

P  P 

I-'-     '-"E^-    •••••  •  • (1> 

1;     P-E-^.'-.  ........    <n> 


h 

47?               ~K 

.      ...     (Ill) 

yi 

Ed           Ed 

E$h 

*     (IV) 

r 

Eh 

(V) 

p 

2  p 

A  pn^h2 

(VI) 

•h 

I 
CPl 

(VII) 

P 
h 

nbth2 
PI           CPl 

.   (VIII) 

128  MACHINE    DESIGN 

Experience  shows  that  thin  plates  have  a  higher  elastic  limit 
than  thick  plates  of  similar  grade  of  material.  In  the  practice  of 
a  prominent  eastern  railway  company,  the  values  allowed  for  the 
maximum  intensity  of  stress  in  flat  steel  springs  are,  for: 

Plates    J    inch  thick,  p  =  90,000  Ibs.  sq.  in. 
T5<r       "       "       ^  =  84,000   " 
|        "       "      ^  =  80,000   "        " 


i        "      "      />  =  75,ooo  " 

The  above  values  are  satisfied  by  the  equation  p  =  60,000  + 
-*-:  —  ,  in  which  h  is  the  thickness  of  plate  in  inches. 

rl 

These  values  are  for  the  greatest  stress  to  which  the  material 
can  be  subjected,  as  when  the  spring  is  deflected  down  against 
the  stops. 

The  modulus  of  elasticity,  E,  may  vary  considerably;  but  its 
value  may  be  assumed  at  about  30,000,000  in  the  absence  of  more 
definite  data. 

In  designing  a  new  spring,  the  value  of  h  is  to  be  found  from 
equation  (III);  then  bl  is  found  by  equation  (VIII).  The  other 
formulae  are  useful  in  checking  springs  already  constructed,  for 
deflection  due  to  a  given  load,  or  the  reverse;  for  safety,  etc. 

43.  Helical  Springs.  If  a  rod  or  wire  be  wound  into  a  flat 
ring  with  the  ends  bent  in  to  the  centre,  Fig.  28,  and  two  equal 
and  opposite  forces,  +  P  and  --  P,  be  applied  to  these  ends 
(perpendicular  to  the  plane  of  the  ring)  as  indicated,  the  rod 
will  be  subjected  to  torsion. 

If  a  longer  rod  be  wound  into  a  helix,  with  the  two  ends  turned 
in  radially  to  the  axis,  the  typical  helical  spring  is  produced.  If 
two  equal  and  opposite  forces,  +  P  and  —  P,  act  on  these  ends, 
along  the  axis  of  the  helix,  they  induce  a  similar  stress  (torsion) 
in  the  rod,  but  as  the  coils  do  not  lie  in  planes  perpendicular  to 
the  line  of  the  forces,  there  is  a  component  of  direct  stress  along 
the  rod.  This  direct  stress  increases  as  the  pitch  of  the  coils  in- 
creases relative  to  their  diameter;  but  with  ordinary  proportions 


SPRINGS  129 

of  springs,  the  torsion  alone  need  be  considered,  when  the  ex- 
ternal forces  lie  along  the  axis  of  the  helix. 

The  following  notation  will  be  used  in  treating  of  helical 
springs  of  circular  wire,  subjected  to  an  axial  load: 

P  =  the  force  acting  along  the  axis. 

r  =  the  radius  of  the  coils,  to  center  of  wire. 

d  =  the  diameter  of  wire. 

^>  =  the  maximum  intensity  of  stress  in  wire  (torsion). 

/p  =  the  polar  moment  of  inertia  of  wire. 

Ea  =  the  transverse  modulus  of  elasticity. 

<5  =  the  ".deflection  "  (elongation  or  shortening)  of  spring. 

n  =  the  number  of  coils  in  the  spring. 

/  =  the  length  of  wire  in  the  helix  =  2  n  r  n  (approximately). 

Suppose  a  helical  spring  under  an  axial  load  to  be  cut  across 
the  wire  at  any  section,  and  the  portion  on  one  side  of  this  section 
to  be  considered  as  a  free  body,  Fig.  29.  Neglecting  the  direct 
stress,  equilibrium  demands  that  the  moment  (Pr)  of  the  external 
force  shall  equal  the  stress  couple,  or  moment  of  resistance 


for  circular  section 


\2  P  d3 

If  this  free  portion  of  the  helix  is  straightened  out,  as  indicated 
by  the  broken  lines  in  Fig.  29,  till  its  direction  is  perpendicular 
to  the  radial  end,  it  will  appear  that  the  moment  Pr  still  equals 

the  moment  of  resistance,  —  p  d3.    Since  the  stress  and  strain  are 

the  same  in  this  helix  and  the  straight  rod,  it  appears  that  the 
energy  expended  against  the  resilience  is  the  same  in  both  cases 
(the  length  of  wire  affected  remaining  constant).  Or,  as  the  force 
(P)  and  the  arm  (r)  are  the  same  in  both  conditions,  the  distances 
through  which  this  force  acts  to  produce  a  given  torsional  stress 
(p)  are  equal.  If  a  straight  rod  of  length  /  is  subjected  to  a 
torsional  moment  Pr,  the  angle  of  twist  being  «  (in  TT  measure)  ,  then 

«/  E 

Pr^-^       •••••.;       . 

[See  Church's  "  Mechanics,"  page  236]. 
9 


130  MACHINE    DESIGN 

The  energy  expended  on  the  rod  is  the  mean  force  applied  mul- 
tiplied by  the  distance  through  which  this  force  acts.  '  If  the  load 
is  gradually  applied,  this  energy  is  >4  P  r  a.  In  the  case  of  the 
corresponding  helical  spring,  the  mean  force  (K-P)  acts  through  a 
distance  equal  to  the  "deflection"  of  the  spring  (fl),  or  the  energy 
expended  is  ^2  P8.  As  pointed  out  above,  the  energy  expended 
in  the  two  cases  is  the  same,  or 

S 
y2  Pro.  =  y2  pd  .-. 


p8 
.  Pr  =  —  JH  =  ^X  -  -  X 


32         2  TT  r  n       64  r2  n 

•'•P  =  H^ •  ••  •  -,  •  •  (l) 

Equation  (i)  may  be  used  for  finding  the  load  corresponding 
to  an  assigned  deflection  in  a  given  spring.  The  equation  can 
be  put  in  the  following  form  for  finding  the  deflection  due  to  a 
given  load : 


•«— /„ 

Or  the  equation  may  be  employed  for  designing  a  spring  in  which 
the  load  and  deflection  are  given,  by  assuming  any  two  of  the 
three  quantities,  r,  d  and  n.  The  most  convenient  form  for  this 
latter  purpose  is  usually, 


These  equations  for  rigidity  hold  good  only  within  the  elastic 
limit  of  the  material,  as  Es  is  simply  a  ratio  between  stress  and 
strain  within  this  limit.  It  therefore  becomes  necessary  to  check 
any  of  the  above  indicated  computations  for  strength,  and  it  will 
often  be  found,  after  thus  checking,  that  the  stress  is  either  too 
high  for  safety,  or  too  low  for  economy. 

The  formula  for  the  strength  of  a  solid  circular-section  rod 
under  torsion  is 


SPRINGS  131 

7T  7T  p  (ft 

Pr  =  —  p  d3  . ' .  P  =  — — : 
16  i6r 

(4) 
*/><P  i6Pr 

'-rfp'>        p"    ^^ 

It  is  to  be  remembered  that  as  equation  (4)  is  for  safe  strength, 
the  load  (P)  should  be  the  maximum  load  to  which  the  spring 
can  be  subjected;  but  equation  (3)  may  be  used  with  any  load 
and  the  corresponding  deflection. 

Example:  The  load  on  a  helical  spring  is  1600  Ibs.,  and  the 
corresponding  deflection  is  to  be  4".  Transverse  modulus  of 
elasticity  of-  material  =  11,000,000,  and  the  maximum  intensity 
of  safe  torsional  stress  =  60,000  Ibs.,  wire  of  circular  section. 
To  design  the  spring,  assume  d  =  W,  and  r  =  iy£";  from  eq.  (3), 

4  X  625  X  11,000,000  X  8 
n  =  -  -  =  10.4. 

4,096  X  64  X  1,600  X  27 

Checking  for  the  stress  by  the  last  equation  in  group  (4), 

16  X  i  ,600  X  1.5  X  512 
p  =  -  —  =  50,200  Ibs. 

-  X  125 

This  stress  is  found  to  be  safe,  but  is  considerably  below  the 
limit  assigned,  and  it  may  be  desirable  to  work  up  to  a  somewhat 
higher  stress.  Another  computation  can  be  made  (with  a  smaller 
d  or  larger  r),  and  by  a  series  of  trials,  the  desired  spring  can 
be  found.  The  following  order  of  procedure  avoids  this  element 
of  uncertainty.  The  load  being  given,  assume  a  diameter  of  wire 
and  value  of  safe  stress,  then  solve  in  eq.  (4)  for  the  radius  of  coil. 
Make  this  radius  some  convenient  dimensions  (not  exceeding  that 
computed  if  the  assumed  stress  is  considered  the  maximum  safe 
value).  Next  substitute  these  values  of  d  and  r  (with  those 
given  for  P,  d  and  E)  in  eq.  (3)  to  find  the  number  of  coils.  Thus, 
with  the  data  of  the  preceding  example,  assuming  d  =  y%" ; 

*   p  d3       TT  X  60,000  X  125  n 

~  16  P      ~  16  X  1,600  X  512 

If  the  $/%"  rod  is  wound  on  an  arbor  3"  diameter,  the  radius  to  the 
centre  of  coils  will  be  about  1.81";  and  the  corresponding  stress 


132  MACHINE    DESIGN 

would  be  60,500  Ibs.  per  square  inch.  This  is  so  slightly  in  ex- 
cess of  the  assigned  value  that  it  may  be  permitted,  especially  as 
this  value  is  a  moderate  one  for  spring  steel.  Substituting  in 

eq.  (3), 

dd*Ea  4  X  11,000,000  X  625 

"  64  P  r3  ~  64  x  i  ,600  X  5-93  X  4,096  ~ 

It  may  be  desirable  to  fix  upon  the  radius  of  coil,  rather  than  the 
diameter  of  wire,  in  the  first  computation,  in  designing  a  spring. 
From  eq.  (4)  : 

-     -     .     (5) 


In  other  cases,  it  may  be  desirable  to  assume  the  ratio  of  the 
radius  of  coil  to  the  diameter  of  wire,  then  from  eq.  (4)  : 


In  either  of  the  preceding  conditions,  a  standard  size  of  wire 
should  be  chosen. 

In  checking  a  given  spring,  it  may  be  required  to  determine 
either  the  safe  load,  or  the  safe  deflection.  If  the  former  is  the 
case,  eq.  (4)  may  be  used  directly.  If  it  is  required  to  find  the 
safe  deflection,  substitute  the  value  of  P  from  eq.  (4)  in  eq.  (2) 
and  the  result  is 


The  weight  of  a  spring  is  a  matter  of  some  importance,  as  the 
material  is  expensive.  The  following  discussion  shows  that  the 
weight  varies  directly  as  the  product  of  the  load  and  the  deflec- 
tion, inversely  as  the  square  of  the  intensity  of  stress  in  the  wire, 
and  directly  as  the  transverse  modulus  of  elasticity.  Hence  for 
a  given  load  and  deflection,  economy  calls  for  a  high  working 
stress  and  a  low  modulus  of  elasticity.  From  eq.  (4)  : 

TT      d3 
P  =  —  p  —  ;  also  for  a  member  under  torsion, 

p  =  ~  X  ^Y  [Church's  "  Mechanics,"  p.  235]. 


SPRINGS  133 

d  *  Es 


P  =  —  x  —  x 

2  TT  r  n        4  TT  r~  n 

(8) 


0     = 


r        2  x  r  n       4  *  r2  n 
4  ~  r2  n  p 


•••'-       .....  .....<„ 

But  the  volume  of  the  spring  is 

v=  y^tfl  =  y2-2d2rn       ......      .      .      .      (10) 


The  weight  is  directly  proportional  to  the  volume;  hence  for 
given  values  of  Es  and  p,  the  weight  varies  simply  as  the  product 
of  the  load  and  the  deflection.  All  possible  helical  springs  (of 
similar  section  of  wire)  have  the  same  weight  for  a  given  load 
and  deflection,  if  of  the  same  material  and  worked  to  the  same 
stress.  It  can  be  shown  that  a  helical  spring  of  square  wire  must 
have  50  per  cent  greater  volume  than  one  of  round  wire,  the 
stress  and  modulus  of  elasticity  being  the  same  in  both.  The 
round  section  is  generally  admitted  to  be  best  for  helical  springs 
under  ordinary  conditions. 

A  small  wire  of  any  given  steel  usually  has  a  higher  elastic 
limit  than  a  larger  one,  while  there  is  not  a  corresponding  change 
in  the  modulus  of  elasticity  with  change  in  diameter.  This 
suggests  the  use  of  as  light  a  wire  as  is  consistent  with  other 
requirements. 

An  extensive  set  of  tests  of  springs,  conducted  by  Mr.  E.  T. 
Adams,  in  the  Sibley  College  Laboratories,  indicates  that  the  steel 
such  as  is  used  in  governor  springs  may  be  subjected  to  stress 
varying  from  about  60,000  Ibs.  per  square  inch  with  ^"  wire  to 
80,000  Ibs.  per  square  inch  (or  more)  in  wire  y%"  diameter.  The 
following  expression  may  be  used  to  find  the  safe  stress  in  such 
springs  : 

1  5  ,000 
^  =  40,000  +  --^  —  .      ....      (12) 

Mr.  J.  W.  Cloud  presented  a  most  valuable  paper  on  Helical 
Springs  before  the  Am.  Society  of  Mechanical  Engineers  (Trans., 


134 


MACHINE    DESIGN 


Vol.  V,  page  173),  in  which  he  shows  that  for  rods  used  in  rail- 
way springs  (%"  to  iT5g-''  diam.),  the  stress  may  be  as  high  as 
80,000  Ibs.  per  square  inch,  and  that  the  transverse  modulus  of 
elasticity  is  about  12,600,000. 

Two  or  more  helical  springs  are  often  used  in  a  concentric  nest 
(the  smaller  inside  the  larger);  all  being  subjected  to  the  same 
deflection.  This  is  common  practice  in  railway  trucks,  where 
the  springs  are  under  compression  when  loaded.  If  these  springs 
have  the  same  "free"  height  (when  not  loaded),  and  if  they 
are  of  equal  height  when  closed  down  "  solid,"  Mr.  Cloud  shows 
that  the  length  of  wire  should  be  the  same  in  each  spring  of  the 
set  for  equal  intensity  of  stress.  The  " solid"  height  of  a  spring 
is  H  =  d  n,  and  the  length  of  wire  is  1  =  2  -  r  n\  hence  the  num- 
bers of  coils  of  the  separate  springs  of  the  set  are  inversely  as  the 
diameters  of  the  wire  and  inversely  as  the  radii  of  the  coils;  or 
the  ratio  of  r  to  d  is  the  same  in  each  spring  of  the  nest.  This 
conclusion  may  be  somewhat  modified  when  it  is  remembered 
that  the  wire  of  smaller  diameter  may  usually  be  subjected  to 
somewhat  higher  working  stress  than  the  larger  wire  of  the  outer 
helices;  and  also  that  the  wire  of  these  compression  springs  is 
commonly  flattened  at  the  end  to  secure  a  better  bearing  against 
the  seats.  See  Fig.  30. 


SUMMARY  OF  HELICAL  SPRING  FORMULAE. 


7T 

Pr  =  — 


(I) 
(II) 


i6P 

d=    I.72.J1.I  (III) 


d=   2.2 

i6Pr 

p  = 

P  =  l67 


(IV) 

(V) 

(VI) 


Ed 


64  r3  n 
64  Pr*n 


n  = 


64  Pr3 

* 


.     (VIII) 

(IX) 

(X) 

(XI) 


SPRINGS  135 

Two  common  methods  of  attaching  "pull"  springs  are  shown 
in  Fig.  30  (a) .  One  end  of  the  spring  shows  a  plug  with  a  screw 
thread  to  fit  the  wire  of  the  spring.  This  plug  is  usually  tapered 
slightly,  and  the  coils  of  the  spring  are  somewhat  enlarged  by 
screwing  it  in.  The  other  end  of  the  spring  shows  the  wire  bent 
inward  to  a  hook  which  lies  along  the  axis  of  the  helix.  The 
former  method  is  usually  preferable  for  heavy  springs. 

Formulae  (I)  to  (VII),  inclusive,  relate  to  strength;  (VIII) 
to  (X),  inclusive,  relate  to  rigidity,  or  elasticity. 

In  the  absence  of  more  exact  information  as  to  the  properties 
of  the  material  of  which  a  steel  helical  spring  is  made,  the  fol- 
lowing values  may  be  taken: 

£.  =  12,000,000, 

15,000 
p  =  40,000  +  — - — . 

44.  Spiral  or  Helical  Springs  in  Torsion.  The  following 
formulae  for  either  true  spiral  or  helical  springs  subjected  to  tor- 
sion are  derived  from  "The  Constructor,"  by  Professor  Reu- 
leaux. 

PRl  PR 

**    ~ET'P'-    ~Z~> 
In  which 

P  =  load  applied  to  rotate  axle, 
R  =  lever  arm  of  this  load, 
V  =  angle  through  which  axle  turns, 
/  =  length  of  effective  coils, 
E  =  modulus  of  elasticity  (direct), 
/  =  moment  of  inertia  of  the  section. 


CHAPTER  VI 
RIVETED  FASTENINGS 

45.  General  Considerations.  The  simplest  form  of  fasten- 
ing is  the  rivet.  It  consists  of  a  head  a  (Fig.  31),  a  straight  shank 
b,  and  a  second  head  c,  which  is  formed  while  hot  and  known 
as  a  point.  When  it  is  desired  to  rivet  two  pieces  M  N  to- 
gether, mating  holes  are  punched  or  drilled  as  shown,  the  rivet 
is  heated  white  hot  and  pushed  into  the  hole  which  is  purposely 
made  a  little  larger  in  diameter.  The  head  is  held  up  firmly 
against  the  plate  by  a  heavy  bar  or  sledge  and  the  point  may  be 
formed  with  a  hand  hammer,  or  with  the  aid  of  a  forming  tool 

or  set.  In  riveting  on  a  large 
scale  this  operation  is  per- 
formed by  hydraulic  or  pneu- 
matic machines.  The  relative 
merits  of  the  two  methods  will 
be  more  apparent  after  further 
discussion.  The  rivet  is  a  per- 
manent fastening  and  cannot 
be  removed  without  the  de- 
FlG  struction  of  either  header  point. 

It  is  largely  used  in  structures 

such  as  bridges,  the  framing  of  buildings,  ship   work,  boilers, 
tanks,  etc. 

Fig.  32  shows  various  forms  of  rivet  heads  and  points.  The 
form  shown  at  B  is  most  commonly  used  for  small  rivets  up  to 
T5e-  in.  diameter,  which  are  driven  without  heating,  for  such  work 
as  light  tank  and  smokestack  work.  The  form  at  C  is  much 
used  in  ship  work,  or  wherever  smooth  exterior  surfaces  are 
desired.  In  machine  work,  where  great  accuracy  is  required, 
the  holes  are  reamed,  and  the  rivet  carefully  fitted  so  as  com- 
pletely to  fill  the  hole;  both  heads  in  such  cases  are  usually 
countersunk  and  formed  cold. 

136 


RIVETED   FASTENINGS  ^y 

When  a  rivet  is  " driven"  hot  it  shrinks  in  cooling,  drawing 
the  riveted  parts  firmly  together.  When  cold  it  is  under  a  ten- 
sile stress  due  to  this  shrinking,  and  for  the  same  reason  it  is 
always  a  little  smaller  than  the  hole  which  it  originally  com- 
pletely filled  when  hot.  The  tensile  stress  due  to  this  cooling 
effect  cannot  be  accurately  determined  as  it  depends  on  the  tem- 
perature of  the  rivet,  and  the  manner  in  which  it  is  driven.  Rivets 
are,  for  this  reason,  unreliable  as  tension  members  and  are  seldom 
so  used.  In  most  cases  the  parts  M  and  N  (Fig.  31)  have  the  load 
P  applied  as  shown,  and  the  tendency  is  to  shear  off  the  rivet 
and  produce  relative  sliding  between  M  and  N.  The  normal 
load,  P',  due  to  the  tensile  stress  in  the  rivet,  holding  the  surfaces 
of  M  and  N  firmly  in  contact,  sets  up  a  frictional  resistance  equal 
to  ftP'  which  opposes  the  action  of  P.  From  experiments  made 


feutton        Con<!>idal  ,        Countersunk 

I     A 


RTHFTIFTl 


FIG.  32. 

by  Stoney  *  it  appears  that  this  frictional  resistance  may  be  taken 
at  about  10,000  Ibs.  per  square  inch  of  rivet  area.  Experiments 
by  Bach,  and  others,  show  a  much  higher  resistance,  but  it  is 
evident  that  if  the  normal  pressure  of  the  rivet  is  such  that  a 
stress  equal  to  or  greater  than  the  elastic  limit  is  induced,  the 
permancy  of  the  resistance  cannot  be  relied  upon. 

In  some  French  and  German  practice  the  design  of  the  joint  is 
based  entirely  upon  the  frictional  resistance,  but  in  England 
and  America  it  is  neglected,  and  the  design  based  upon  the 
tensile  and  shearing  strength  of  the  plates  and  rivets. 

46.  Forms  of  Joints.  Riveted  joints  are  of  many  forms  de- 
pending on  the  character  of  the  work  to  which  they  are  applied. 
In  structural  work,  such  as  bridges,  they  are  used  simply  to 

*  "Strength  of  Riveted  Joints,"  page  75. 


138  MACHINE    DESIGN 

resist  direct  loads;  but  in  boiler  construction,  and  similar  work, 
they  must  not  only  resist  direct  loading  but  must  also  be  tight 
against  fluid  pressure.  This  last  requirement  materially  affects 
the  proportions  of  the  joint,  and  makes  the  design  of  joints  for 
withstanding  fluid  pressure  most  important.  Riveted  joints  are 
divided  into  two  general  forms. 

(a)  Lap  Joints,  where  the  sheets  to  be  joined  are  lapped  on 
each  other  and  riveted  as  shown  in  Fig.  33  (a). 

(b)  Butt  Joints,  where  the  edges  of  the  sheets  to  be  joined 
abut  against  each  other,  anoV  have  auxiliary  butt  straps  or  cover 
plates  riveted  to  the  edge  of  each,  as  shown  in  Fig.  33  (e)  and  (f ) . 

A  lap  joint  may  have  one  or  more  rows  or  seams  of  rivets, 
and  these  rows  way  be  arranged  in  the  form  of  "chain"  rivet- 
ing, Fig.  33  (b),  or  in  the  form  of  zigzag  or  staggered  riveting, 

Fig.  33  (c). 

A  butt  joint  may  have  one  or  more  seams  of  rivets  on  each 
side  of  the  joint,  and  these  may  also  be  arranged  in  either  chain 
or  staggered  form,  as  shown  in  Fig.  33  (f)  and  (g). 

The  combinations  that  may  be  thus  made  up  are  very  numer- 
ous, and  the  student  is  referred  to  any  treatise  on  boiler  work  for 
fuller  information  on  this  point. 

The  distance  between  rivets  along  the  seam  is  called  the  pitch 
or  spacing,  and  will  be  denoted  by  s,  Fig.  33  (b).  An  examina- 
tion of  any  riveted  joint  will  show  that  the  arrangement  of 
rivets,  or  pattern  as  it  may  be  called,  continually  repeats  itself  as 
the  "seam"  extends  along  the  joint,  the  repetition  occurring  with 
the  greatest  pitch,  where  the  pitch  of  the  various  seams  is  unequal 
as  in  Fig.  33  (h) .  A  unit  strip  is  equal  in  width  to  the  pitch,  the 
maximum  pitch  being  taken  when  the  pitch  of  all  seams  is  not 
the  same.  The  transverse  pitch  is  the  distance  between  the 
centre  lines  of  adjacent  seams  Fig.  33  (b)  and  will  be  denoted 
by  st.  The  diagonal  pitch  is  the  distance  between  the  centre  of 
a  rivet  and  that  of  the  one  nearest  to  it.  diagonally,  in  the  next 
row,  and  will  be  denoted  by  sd  Fig.  33  (d).  The  margin  is  the 
distance  from  the  edge  of  the  plate  to  the  center  line  of  the  nearest 
row  of  rivets,  as  e  Fig.  33  (d).  It  is  sometimes  denned  as  the 
distance  from  the  edge  of  the  plate  to  the  edge  of  the  rivet  hole. 


RIVETED   FASTENINGS 


139 


47.  Stresses  in  Riveted  Joints.  The  stresses  that  exist  in 
the  various  members  of  riveted  joints  are  complex,  and  do  not 
admit  of  refined  calculation.  Not  only  are  the  plates  subjected 
to  the  apparent  direct  stresses  of  tension  and  compression,  and 


(*) 


. 
cit       dit        ID 

\       i  r~~1    l    .      I    r~~l  ;  r~7 


FIG.  33. 
(a,  b,  c,  d,  e,  f,  g,  h.) 


140 


MACHINE    DESIGN 


the  rivets  to  shear  and  compression,  but  often  there  are  also 
bending  actions  which  are  difficult  to  analyze  and  provide  for 
mathematically.  Thus  a  simple  lap  joint,  as  that  shown  in  Fig. 
33  (a),  when  subjected  to  a  load,  tends  to  take  the  form  shown 
in  Fig.  34.  The  force  applied  tends  to  draw  the  plates  into  the 
same  plane,  Cutting  a  bending  action  on  the  plate  and  rivet,  a 
greater  tensile  stress  on  the  rivet  head,  and  a  concentrated  crushing 
load  on  the  corners  of  the  sheets.  The  frictional  resistance  is 
entirely  destroyed  when  the  conditions  illustrated  in  Fig.  34  exist. 

The  above  defects  are  more  marked  in  the  lap  joint  than  in 
the  double  strapped  butt  joint,  as  in  the  latter  the  plates  are  ini- 


(a) 


(e) 


tially  in  line  and  the  condition  shown  in  Fig.  34  cannot  occur.  But 
even  here  the  rivets  do  not  completely  fill  the  holes  when  cold,  and 
hence  some  bending  of  the  rivet  and  concentrated  crushing  on  the 
plate  must  result.  Again,  while  the  quality  of  the  material 
forming  the  joint  may  be  well  known  or  determined,  the  work- 
manship is  not  so  easily  controlled  and  may  be  very  defective 
and  yet  not  show  on  the  exterior;  and  while  there  have  been 
many  tests  *  made  to  find  the  ultimate  strength  of  riveted  joints, 

^Proceedings   Institute   of   Mechanical   Engineers,    1881,    1882,    1885,    1888. 
Watertown  Arsenal  Reports,  1885,  1886,  1887,  1891,  1895,  1896. 


RIVETED   FASTENINGS  141 

such  tests  show  only  the  stress  at  which  a  certain  element  of  the 
joint  failed,  and  do  not  throw  any  light  on  the  distribution  and 
progress  of  the  stresses  in  the  various  individual  members  dur- 
ing the  test.  Such  tests  have  usually  been  performed  on  joints 
made  of  straight  plates  while  in  practice  these  are  often  curved. 
These  experiments,  therefore,  while  giving  the  only  data  available 
relative  to  the  ultimate  strength,  should  be  used  with  judgment 
in  designing.  For  these  reasons  the  theoretical  formulas  deduced 
for  the  design  of  riveted  joints,  as  a  rule,  take  cognizance  only 
of  the  apparent  simple  stresses  and  provide  for  the  unknown  by 
means  of  a  factor  of  safety. 

It  has  been  found  that  riveted  joints  may  fail  in  one  of  the 
following  ways : 

(a)  Shearing  of  the  rivet  as  in  Fig.  35  (a). 

(b)  Rupturing  of  the  plate  by  tension  as  in  Fig.  35  (b). 

(c)  Tearing  of  the  margin  as  in  Fig.  35  (c). 

(d)  Shearing  of  the  margin  as  in  Fig.  35  (e). 

(e)  Crushing  the  plate,  or  rivet  as  in  Fig.  35  (d). 

(f)  Rupturing  of  the  plate  diagonally  between  rivet  holes  by 
tension,  in  staggered  riveting. 

Where  the  joint  is  complex  in  form,  ultimate  failure  may  be 
due  to  one  or  more  of  the  above  causes.  The  Watertown  Arsenal 
reports  include  cuts  of  ruptured  joints  which  are  very  instructive 
on  this  point.  Figures  (a)  and  (b),  on  Plate  I,  are  reproduced 
from  these  reports,  and  show  very  clearly  all  the  ways  in  which 
failure  may  occur  in  the  plate.  Fig.  (c)  shows  a  rivet  that  has 
been  tested  to  destruction  in  single  shear,  while  Fig.  (d)  shows 
one  that  has  been  similarly  tested  in  double  shear. 

It  is  obvious  that  no  riveted  joint  can  be  as  strong  as  the  un- 
perforated  plate,  since  the  very  fact  of  making  holes  in  it  reduces 
the  cross-sectional  area  in  the  line  of  the  rivet  holes.  The  ratio 
of  the  strength  of  the  weakest  element  of  the  joint,  to  the  strength 
of  the  unperforated  plate,  is  called  the  relative  strength  or  ef- 
ficiency of  the  joint.  The  first  expression  is  more  suggestive 
and  will  be  used  in  this  work.  It  is  desirable  to  reduce  the 
strength  of  the  plates  as  little  as  possible  by  perforation;  and  if, 


PLATE  I. 


RIVETED   FASTENINGS 


143 


therefore,  the  correct  relation  between  the  size  of  rivet  and  cross 
section  of  perforated  plate,  for  equal  strength,  is  established,  an 
excess  of  strength  in  other  directions,  as  marginal  distance,  is 
not  a  defect  but  good  design,  as  it  insures  that  the  full  strength 
of  the  perforated  plate  will  be  in  service  before  rupture  can  occur. 
A  well-designed  joint  should  hence  fail  by  tearing  of  the  sheet 
along  the  line  of  the  rivet  holes,  at  about  the  same  load  as  will 
destroy  the  rivets;  and  the  relative  strength  of  a  well-designed 
joint  should  be  the  ratio  of  the  cross  section  of  the  perforated  to 
that  of  the  unperforated  plate,  the  shearing  and  crushing  resist- 
ance of  the  rivets  being  equal  to  the  former.  If  this  equality  does 
not  exist,  the  relative  strength  of  the  joint  can  be  made  greater 
by  strengthening  the  weaker  of  these  elements  at  the  expense  of 
the  stronger. 

48.  Marginal  Strength.     The  width  of  margin  is  independ- 
ent of  the  proportions  of  the  other  elements,  and  hence  can  be 
made  sufficient  to  prevent  tearing  or  shearing,  as  in  Fig.  35  (c) 
and  (e).     It  has  been  found  that,  with  the  usual  proportions,  if 
the  margin  be  made  equal  to  one  and  one  half  the  diameter  of 
the  rivet,  it  will  be  safe  against  both  shearing  and  tearing  from 
rivet  pressure.     A  Committee  of  the  Master  Steam  Boiler  Makers 
Association  recently  recommended,  as  a  result  of  experiments, 
that  the  distance  from  the  center  of  the  rivet  to  the  edge  of  the 
plate  be  made  twice  the  diameter  of  the  rivet,  in  order  to  insure 
excess  strength  enough  to  either  shear  the  rivets  or  rupture  the 
plate  by  tension.    It  is  important,  however,  that  the  margin  be  not 
excessive  in  boiler  work  as  this  makes  it  more  difficult  to  make  a 
steam-tight  joint  by  calking  the  edge  of  the  plate.     The  con- 
sideration of  the  marginal  strength  can  hence  be  omitted  as  far 
as  its  influence  on  the  relative  strength  of  the  joint  is  concerned, 
as  it  can  be  made  to  depend  on  the  diameter  of  the  rivet. 

49.  Transverse    and    Diagonal    Pitch.     It    has    been    deter- 
mined, in  a  similar  way  as  above,  that  in  chain  riveting  the 
transverse  pitch  should  not  be  less  than  twice  the  diameter  of  the 
rivet  or  2d,  where  d  is  the  diameter  of  the  rivet,  and  2.5  d  is 
better.     It    has    also   been   demonstrated    mathematically    (see 
Cathcart's  "  Machine  Design,"  page  148)  that  in  staggered  riveting 


144  MACHINE    DESIGN 

the  transverse  pitch  should  not  be  less  than  0.4  times  the  pitch 
along  the  seam,  in  order  to  avoid  rupture  along  the  diagonal 
pitch,  and  a  greater  distance  is  recommended  for  safety.  Unwin 
(page  123)  gives  2  d  as  the  minimum  diagonal  pitch  in  staggered 
riveting,  which  would  make  the  transverse  pitch  1.7  d,  and 
recommends  that  somewhat  greater  distances  be  used  for  added 
strength.  An  examination  of  the  practice  of  several  boiler-mak- 
ing and  insurance  concerns  shows  that  these  values  check  fairly 
well  with  practice.  It  appears  from  the  above  that  the  trans- 
verse pitch  can  also  be  made  to  depend  on  the  diameter  of  the 
rivet,  though  it  is  not  a  direct  function  of  the  rivet  diameter. 

50.  Theoretical  Strength  of  Riveted  Joints.  Since  the  margin 
and  transverse  pitch  can  be  assigned  from  the  diameter  of  the 
rivet,  three  of  the  ways  in  which  a  joint  may  fail,  namely  c,  d  and 
/,  page  141,  can  be  omitted  from  the  theoretical  discussion  of  the 
strength  of  riveted  joints,  leaving  a,  b  and  e  to  be  considered; 
the  problem  being  so  to  proportion  the  rivet  and  the  pitch  along 
the  seam  as  to  give  equal  strength  against  failure  in  any  of  these 
three  ways.  Let, — 

d  =  diameter  of  rivet  in  inches. 

s  =  pitch  of  rivets  in  inches. 

pt  =  tensile  strength  of  plates  in  pounds  per  sq.  inch. 

pc  =  crushing  strength  of  plates  or  rivets  in  pounds  per  sq. 

inch,  if  rivets  are  in  single  shear. 
p'c  =  crushing   strength   of  plates  or  rivets  in   pounds   per 

inch,  where  rivets  are  in  double  shear. 
ps  =  shearing   strength   of  rivets   in   pounds   per   sq.    inch, 

when  in  single  shear. 
p'e  =  shearing  strength  of  rivets  in   pounds  per    sq.   inch, 

when  in  double  sheaf. 

It  is  known  that  the  unit  shearing  resistance  of  a  rivet  is 

greater  in  single  shear  than  in  double  shear,   while  the  unit 

crushing  resistance  is  less  in  single  shear  than  in  double  shear. 

Consider  first  a  simple  lap  joint  (see  Fig.  33  a).     The  tensile 

strength  of  the  unperforated  strip  is 

P  =  stpt        ........      (i) 


RIVETED   FASTENINGS  145 

The  tensile  strength  of  the  perforated  strip  along  the  seam  of 
rivets  is 

T=  (s-d)tpt    .      .      .      .      .      .      (2) 

In  the  simple  lap  joint  there  is  but  one  rivet  per  unit  strip  and 
its  shearing  strength  is, 

7T<f 

—  P.      '  •      (3) 

The  resistance  to  crushing  of  the  rivet  or  the  plate  against  which 
it  bears  is, 


For  uniform  strength  against  rupture  and  hence  for  greatest 
relative  strength, 

T  =  S  =C 
Equating  (3)  and  (4) 

xd2 

—  A-  "A 

.••*--7<J; 

Equating  (2)  and  (3) 


T        (s  —  d)tpt      s—d 
The  relative  strength  =  —  =  -       -£—£-«  =  - 

r  S  t  Pt  S 

Double-Riveted  Lap  Joints.  In  a  similar  way  equations 
may  be  developed  for  any  other  form  of  joint.  Thus  for  double- 
riveted  lap  joints 


and 


*  It  is  known  from  experiments  on  indentation  that  the  resistance  to  indentation 
depends  very  little  on  the  form  of  the  indenting  body  but  mainly  on  its  projected 
area.    Hence  it  is  customary  to  take  the  resistance  of  rivets  to  crushing  as  propor- 
tional to  their  projected  area. 
10 


146  MACHINE    DESIGN 

The  relative  strength  =  -      -  and  will  be  greater  than  in  the  case 

of  the  single-riveted  lap    joint  since  s  is  greater  in  proportion 
tod. 

Single-Riveted  Butt  Joints  with  double  cover  plates. 
Here, 


and  the  relative  strength  =  - 

Double-Riveted  Butt  Joints  with  either  chain  or  staggered 
riveting  and  double  cover  plate. 
Here, 


Note  that  the  relative  strength  =  -    —  ,  and  compare  the  rel- 

ative values  of  s  and  d  in  this  case  with  those  in  single  riveting. 
Thickness  of  Cover  Plates.  It  is  evident  that  where  only 
one  cover  plate  is  used  its  thickness  should  not  be  less  than  that 
of  the  main  plate;  and  in  practice,  single  cover  plates  are  made  a 
little  thicker  than  the  main  plate  to  insure  an  excess  of  strength. 
A  butt  joint  with  a  single  cover  plate  is  shown  at  Fig.  33  (e).  A 
joint  of  this  kind  is  really  equivalent  to  two  lap  joints.  They 
are  used  where  a  smooth  surface  is  desired  or  in  such  places  as 
the  longitudinal  seams  of  steam  boilers  where  a  lap  joint  has 
sufficient  strength.  Double  butt  straps  should  not  be  made  less 
than  half  the  thickness  of  the  main  plate,  and,  for  the  same 
reason  as  above,  it  is  not  unusual  to  increase  their  thickness  to 
about  T8^  t  where  the  cover  plates  are  the  same  width.  Where 
the  outer  cover  plate  is  narrower  than  the  inner  plate,  as  in 


RIVETED    FASTENINGS  147 

Fig.  33  00,  the  outer  cover  plate  is  often  of  the  same  thickness  as 
the  main  plate  and  the  inner  one  from  ^  to  T%  /. 

51.  General  Equations  for  Riveted  Joints.  The  funda- 
mental equations  for  riveted  joints  may  be  put  in  a  more  general 
form.  The  unit  strip,  as  before,  is  of  width  equal  to  the  pitch; 
the  maximum  pitch  being  taken  for  such  width  of  unit  strip  if 
all  rows  do  not  have  the  same  pitch. 

Let  kl  =  pc  -r-  ps-,  k2  =  p'c  +  p's\  n  =  number  of  rivets  per 
unit  strip  in  single  shear  and  m  =  number  of  rivets  per  unit 
strip  in  double  shear  per  unit  strip  of  joint. 

The  general  expression  for  the  net  tensile  strength  of  the  unit 
strip  is 

T  =  (S  -  d)  t  A  *      .     .     .     .     ,     (i) 

The  general  expression  for  resistance  to  shearing  of  the  rivets 
in  the  unit  strip  is 

n*  tf  2  m* d? 

s  =  —  >.  +  -— -f.  « 

The  general  expression  for  resistance  to  crushing  of  the  rivets 
in  the  unit  strip  is 

C  =  nd-tkip't  +  mdtk2p's      .      .      .      ,     (3) 
The  tensile  resistance  of  the  solid  strip  is 

P  =  stp ,.      .     (4) 

Equating  S  and  C,  eqs.  (2)  and  (3)  and  solving  for  d 

±      nklP.  +  mktp: 

n  ps  +  2  m  pa' 

which  gives  the  proper  diameter  of  rivets  for  a  given  thickness  of 
plate,  when  the  number  of  rivets  in  single  shear  and  the  number 
in  double  shear  and  the  corresponding  shearing  and  crushing 
resistances  are  known. 

*  Where  the  rows  of  rivets  do  not  all  have  the  same  pitch,  as  in  some  forms  of 
butt  joints,  the  outer  row  or  that  farthest  from  the  edge  of  the  sheet  has  the  greatest 
pitch  (see  Fig.  33  h).  It  is  evident  that  if  the  sheet  yield  at  all  by  tearing,  it  will 
yield  along  this  outer  row  of  rivets;  for  it  cannot  tear  along  an  inner  row  without 
shearing  the  outer  row  of  rivets,  and  it  cannot  shear  one  row  of  rivets  without  shear- 
ing all,  in  which  case  the  joint  would  yield  by  shearing  of  the  rivets  and  not  by 
tearing. 


148  MACHINE    DESIGN 

Equating  T  and  5,  eqs.  (i)  and  (2),  and  solving  for  s 


Equating  S  and  C,  eqs.  (2)  and  (3) 

-  —  (up  +  2  m  ps')  =  d  t  (n  k,  p^  +  m  k2  pf) 
4 

*<P(np.+  2m 

= 


Equating  T  and  S,  eqs.  (i)  and  (2)  and  solving 

stp=^(nps+  2W#.')  +  dtpt  =  P.      >;.     (8) 
4 

If  the  joint  is  designed  for  maximum  relative  strength,  T^ 
S=C,  hence  any  one  of  these  three  quantities  divided  by  P 
gives  the  relative  strength  (E)  of  the  ideal  joint,  for  any  given 
form,  or  dividing  (2)  by  (8) 


p  ~~  ~<f 

-  (n  ps  +  2  m  ps'}  +  dtp, 

Substituting  the  value  of  d  t  as  given  by  eq.  (7)  and  dividing 

numerator  and  denominator  by (n  ps  +  2  m  ps'), 

4 


E  =  -  -^  --  •  -•  --r—  .    (9) 

•w 

nk^p.  +  mki  p,'  ^  £>C+AW  £c 

If  all  rivets  are  in  single  shear, 

E  =  -  -i-r-       .....     (90 

I  +  ^T7 

n  kt  ps 


t    s 
If  all  the  rivets  are  in  double  shear, 

E  =  -    -^—  .    .  ~      (  •  *~     (9") 


RIVETED   FASTENINGS  149 

Equation  (9)  applies  to  any  form  of  riveted  joint.  It  is 
useful  in  finding  the  limiting  relative  strength  of  joint  for  any 
form  and  materials;  the  actual  proportions  adopted  may  give  a 
lower  relative  strength,  but  can  never  give  higher  relative  strength. 

These  general  equations  were  originally  due  to  Professor 
William  N.  Barnard,  who  also  suggested  the  above  expressions 
for  the  maximum  relative  strength  in  the  general  case. 

The  forms  in  which  eqs.  (9),  (9')  and  (9)"  are  now  given  are 
due  to  Professor  H.  F.  Moore. 

The  following  are  rough  average  values  of  the  relative  strength 
of  joints  as  made  in  practice  for  boiler  work: 

Single  riveted  lap  joints 55 

Double  riveted  lap  joints 70 

Single  riveted  butt  joints 65 

Double  riveted  butt  joints 75 

Triple  riveted  butt  joints ' 80 

Quadruple  riveted  butt  joints 85 

52.  Practical  Considerations  Affecting  Proportions  of  Riveted 
Joints.  It  is  to  be  especially  noted  that  the  proportions  of 
riveted  joints  as  given  by  the  foregoing  equations  are  based 
on  equal  strength  of  rivet  and  plate,  and  that  any  variation 
therefrom  will  destroy  this  theoretical  equality.  It  is  apparent 
also  that  any  variation  in  the  strength  of  the  material  used  would 
affect  the  proportions  as  given  by  these  equations,  and  that  a 
table  of  rivet  diameters  and  pitches  would  have  to  be  very  ex- 
tensive to  cover  the  entire  range  of  practice.  It  is  an  advantage 
in  practice,  however,  to  adopt  regular  diameters  and  pitches  for  a 
given  thickness  of  plate  and  form  of  joint.  It  has  also  been 
found  that  as  the  thickness  of  the  plate  increases,  the  correspond- 
ing theoretical  diameter  of  the  rivet  sometimes  becomes  too  large 
to  be  easily  driven,  especially  in  the  case  of  simple  lap  joints. 
In  the  case  of  boilers,  or  wherever  fluid  pressure  must  be  with- 
stood, the  theoretical  spacing  must  sometimes  be  modified  in 
order  that  it  may  not  be  so  great  as  to  prevent  the  making  of  a 
steam-tight  joint.  Wherever  such  variations  are  made  the  general 
expressions  for  T,  S  and  C  can  always  be  used  to  check  the 
strength  and  show  in  what  direction  the  joint  may  be  strength- 


150  MACHINE    DESIGN 

ened,  with  the  fundamental  object  of  making  it  strong  enough 
in  all  other  directions  to  insure  full  service  out  of  the  plate  itself. 
It  may  be  noted  that  the  bearing  resistance  of  a  rivet  varies  with 
the  diameter,  while  the  shearing  resistance  varies  with  the  square 
of  the  diameter.  If,  therefore,  the  rivet  chosen  be  smaller  in 
diameter  than  would  be  given  by  the  theoretical  equations,  the 
shearing  resistance  alone  need  be  regarded ;  while  if  the  diameter 
of  the  rivet  be  greater  than  the  theoretical  diameter,  the  bearing 
pressure  only  need  be  considered. 

If  the  joint  to  be  made  does  not  have  to  withstand  fluid  or 
gaseous  pressure,  the  design  can,  for  ordinary  thickness  of 
plate,  be  made  to  conform  closely  to  the  theoretical  proportions 
for  equal  strength ;  but  when  fluid  or  gaseous  pressure  must  be 
withstood,  as  in  boiler  work,  the  spacing  of  the  rivets  for  thick 
plates  must  be  less  than  the  theoretical  spacing  to  insure  tight- 
ness; and  in  all  cases  as  the  plates  increase  in  thickness,  the 
diameter  of  the  rivet,  as  already  noted,  is  for  practical  reasons 
reduced  in  diameter  from  that  required  for  equal  strength.  The 
relation  between  diameter  of  rivet  and  thickness  of  plate  as  fixed 
by  average  practice  may  be  expressed  by  the  equation, 

d  =  1.2  \/ 1 (10) 

For  plates  above  y%"  thick  this  equation  will  give  rivets  smaller 
in  diameter  than  required  for  equal  strength  in  all  directions. 
As  before  pointed  out,  the  rivets  in  such  cases  need  only  be 
checked  for  shearing  strength.  If  the  diameter  of  the  rivet  be 
determined  by  equation  (10),  and  the  pitch  so  chosen  as  to  make 
the  tensile  strength  of  the  perforated  plate  equal  to  the  shearing 
strength  of  the  rivet,  the  maximum  relative  strength  of  joint 
possible  with  the  rivet  chosen  will  be'  obtained,  and  it  will  be 
found  that  the  joints  will  be  steam-tight. 

Example.  It  is  required  to  design  a  riveted  joint,  as  shown 
in  Fig.  33  (h) ,  for  a  boiler  shell  in  which  the  force  tending  to  pull 
the  joint  apart  is  6,000  pounds  per  inch  of  length  of  shell.  The 
plate  is  to  be  of  steel  of  60,000  Ibs.  tensile  strength,  the  rivets 
are  to  be  of  steel  and  are  to  have  a  shearing  strength  of  49,000 
Ibs.  per  square  inch  in  single  shear,  and  42,000  Ibs.  per  square 
inch  in  double  shear.  The  factor  of  safety  is  to  be  5. 


RIVETED   FASTENINGS  151 

The  allowable  stress  per  inch  of  length  of  the  shell  outside 

f\f\  P)OO 

the  joint  is  -         -  =  12,000  Ibs.     If  the  joint  were  as  strong  as 

the   imperforated    plate   the   thickness   of    the   plate   would    be 

6,000 
=  X  inch.       The  relative  strength  of  the  joint  will  not 

be  less  than  «8o,  and  hence  the  thickness  of  the  plate  must  be 

"A=^L 
.80      8  ' 

j—      7" 
The  diameter  of  the  rivets  will  =  1.2  A|  A  =  I—  nearly. 

The  size  of  the  punched  hole  and  the  diameter  of  the  driven 
rivet  will  be  ff".  Equation  (6)  of  this  chapter  gives  the  rela- 
tion between  pitch  and  diameter  of  rivet  for  equal  strength 
against  shearing  of  the  rivets  and  tearing  of  the  plate. 

~d?rnp,  +  2ra/?/~| 
Thus,  5  -  -          Fs   ,         ?°      +  d 
4  L          tpt          -I 

Here,  n  =  i  and  m  =  4. 

Hence,  s  -  ^^^000^(2,^^000^       tf 

4      L  5/s  X  60,000 

.*.  s  =  8  inches  nearly. 

The  relative  strength  of  the  joint  is 

g 15 

^  =  88%,  hence  the  design  is  safe. 

o  t 

If  the  pitch  found  as  above  should  be  considered  too  great, 
either  on  account  of  very  high  steam  pressure  or  because  it  is 
desired  to  make  the  structure  stiffer,  a  smaller  pitch  could  be 
used,  but  the  relative  strength  would  be  less. 

Where  no  fluid  pressure  is  to  be  withstood  the  above  methods 
will  always  give  satisfactory  results  for  joints  in  tension.  For 
joints  in  compression  the  student  is  referred  to  treatises  on  struc- 
tural work. 

52.1.  The  Making  of  Riveted  Joints.  It  is  evident  that  the 
following  precautions  must  be  observed  in  making  first-class 
riveted  joints. 


152  MACHINE    DESIGN 

(a)  The  plates  must  be  in  close  contact  before  the  rivet  is 
driven,  to  prevent  a  fin  from  forming  between  them  and  thus 
making  a  tight  joint  impossible. 

(b)  The  mating  holes  must  be  "fair";  that  is,  they  must  be  in 
perfect  alignment  to  insure  full  cross  section  of  the  rivet  at  the 
junction  of  the  plates. 

(c)  The  rivet  must  completely  fill  the  hole. 

(d)  The  rivet  should  be  carefully  driven  so  that  its  strength, 
or  that  of  the  plate,  will  not  be  weakened  by  poor  workmanship. 

(a)  In  hand  riveting  the  plates  are  drawn  up  together,  before 
the  rivet  is  driven,  by  a  bolt  placed  in  a  hole  near  that  in  which 
the  rivet  is  to  be  driven.     With  comparatively  thin  plates  this 
method  will  accomplish  the  result  if  the  holes  have  been  accu- 
rately spaced,  and  if  the  plates  have  been  properly  rolled  and  fit 
well.     For  heavy  work,  where  for  other  reasons  machine  riveting 
is  necessary,  the  riveter  is  sometimes  provided  with  a  power- 
driven  closing  device  which  holds  the  plates  up  till  the  rivet  nips 
from  cooling. 

(b)  The  rivet  holes  in  the  plate  may  be  either  punched  or 
drilled.     They  are  generally  made  about  iV  inch  larger  than  the 
rivet.     Generally  speaking  it  is  cheaper  to  punch  the  holes  than 
to  drill  them,  and  hence  in  the  cheaper  kinds  of  work,  and  with 
thin  plates,  punching  is  almost  always  resorted  to.     In  structural 
work  the  holes  are  generally  punched.     There  are,  however,  some 
serious   objections   to   punching.     When   the   punch   is   forced 
through  a  plate  the  amount  of  metal  which  it  removes  in  the  form 
of  a  "plug"  is  not  equal  to  the  amount  that  originally  filled  the 
hole.     This  is  accounted  for  by  the  fact  that  punching  is  not  a 
pure  shearing  action,  but  that  during  the  process  there  is  a  flow 
of  metal  from  under  the  punch  to  the  walls  of  the  hole,  setting 
up  a  stress  in  the  material,  and,  if  the  metal  is  at  all  hard,  seri- 
ously impairing  the  strength  of  the  plate.     It  is  found  that  this 
action  is  confined  to  a  thin  ring  next  to  the  hole,  and  that  by 
either  reaming  out  the  hole  about  yV  inch  all  around,  or  anneal- 
ing the  plate,  this  weakening  effect  disappears.     The  process  of 
punching  is  apt  to  make  inaccurate  work  and,  therefore,  when 
the  plates  are  brought  together  the  mating  holes  are  not  fair.     The 


RIVETED    FASTENINGS  153 

old  practice  of  driving  a  taper  drift  pin  into  such  holes  and 
drawing  them  into  line  by  force  is  now  largely  prohibited, 
the  injury  thus  done  to  the  plate  being  often  very  serious.  If, 
however,  the  holes  are  punched  a  little  small,  and  put  together 
and  reamed  to  the  proper  size,  the  difficulties  due  to  punching 
are  largely  overcome.  Thin  plates,  in  which  the  effect  of  punch- 
ing is  small,  are  punched  and  used  without  reaming  or  annealing. 
Plates  more  than  X  inch  thick  should  always  be  either  annealed 
or  have  the  holes  reamed  after  punching.  Heavy  plates  are 
always  better  if  drilled,  and  all  first-class  boiler  work  requires 
the  holes  to  be  drilled  in  place  and  all  burrs  carefully  removed. 
This  last  is  important,  as  the  burr,  if  allowed  to  remain,  may 
seriously  impair  the  strength  of  the  head.  A  small  countersink, 
on  the  other  hand,  materially  contributes  to  the  strength  of  the 
rivet.  When  plates  are  annealed  the  work  should  be  properly 
done;  for  if  the  plate  be  overheated  structural  changes  take  place 
that  materially  weaken  the  metal.  The  heating  should  not  be 
too  rapid  nor  the  temperature  above  a  medium  cherry  red.  The 
holes  made  by  punching  are  necessarily  somewhat  tapering  in 
form,  and  where  they  are  used  as  they  come  from  the  punching 
machine  the  rivet  holes  should  be  punched  so  that  they  will  come 
together  as  in  Fig.  36;  for  the  rivet  drives  better  and  the  tapering 
rivet  has  a  tendency  to  relieve  the  head  of  part  of  the  tensile  stress. 

(c)  Since  it  is  necessary  to  have  the  hole  a  little  larger  than 
the  rivet,  it  is  clear  that  the  rivet  when  driven  must  be  upset 
throughout  its  entire  length  in  order  that  it  may  completely  fill 
the  hole.     Large  rivets  should  therefore  be  machine  driven,  as  it  is 
difficult  to  upset  heavy  rivets,  especially  if  of  great  length,  by 
hand.     If  a  rivet  does  not  completely  fill  the  hole  an  undue  con- 
centrated bearing,   or  shearing  stress,   may  be  brought  on  its 
neighbor. 

(d)  On  the  other  hand,  care  must  be  exercised  in  machine 
riveting  that  the  pressure  applied  does  not  create  such  a  flow  in 
the  rivet  as  unduly  to  strain  the  plate,  and  also  that  the  pres- 
sure applied  is  not  great  enough  to  crush  the  plate  directly. 
Practice  allows  about  80  tons  per  square  inch  of  rivet  area. 
Machine  riveting,  when  well  done,  is  superior  to  hand  work, 


154  MACHINE    DESIGN 

the  plates  being  held  up  firmer,  and  also  because  the  impact 
from  hand  riveting,  especially  if  the  rivet  is  worked  too  cold,  is 
liable  to  result  in  the  breaking  off  of  the  head.  In  either  case 
care  should  be  exercised  that  the  point  is  formed  on  the  rivet 
concentrically;  if  the  dies  are  not  properly  set  eccentric  rivet- 
ing will  occur.  In  machine  riveting  the  pressure  should  be  re- 
tained on  the  rivet  till  cold  enough  to  hold  the  plate  firmly. 
This  is  sometimes  recognized  in  writing  specifications. 

If  the  spacing  of  the  rivet  is  correct  and  the  riveting  well 
done,  the  joint  will  be  tight  against  ordinary  pressures.  Where 
tight  joints  do  not  result  the  edge  of  the  plate  is  "  calked."  This 
is  often  done,  as  shown  in  Fig.  36,  by  means  of  a  sharp-nosed 
tool  T  which  tucks  the  sharp  bevelled  edge  of  the  plate  under- 
neath, as  shown  in  an  exaggerated  manner.  There  is  liability 
of  injuring  the  lower  plate  in  using  a  sharp-nosed  tool  as  T,  and 
the  method  shown  at  B,  Fig.  36,  is  preferable.  The  plates  should 
be  bevelled  before  riveting,  as  the  method  of  hand  bevelling  after 
riveting,  as  often  done  in  practice,  is  almost  sure  to  result  in  some 
injury  to  the  lower  plate. 

52.2.  Strength  of  Materials  for  Riveted  Joints.  It  is  well 
known  that  the  strength  of  rivets  is  different  in  single  and  double 
shear.  The  following  may  be  used  as  average  values. 

PS*  pet 

Iron  rivets  single  shear 40,000  60,000 

Iron     "      double    "  39,ooo  72,000 

Steel    "       single      "   49,000  80,000 

Steel    "      double    "    42,000  100,000 

Steel  Plates  for  Boiler  Work  are  generally  specified  to  have 
a  tensile  strength  of  not  less  than  55,000  Ibs.  per  square  inch, 
and  not  more  than  65,000  Ibs.  per  square  inch,  for  if  the  tensile 
strength  is  too  high  and  the  metal  is  hard  they  are  liable  to 
crack  while  being  worked.  For  structural  steel  construction 
the  student  is  referred  to  handbooks  on  structural  work.  For 
iron  plates  an  average  value  may  be  taken  as  45,000  pounds  per 

*  Master  Boiler  Makers'  Association  Rules,  page  150. 

t  Proceedings    Inst.    Mech.   Engineers,    1885,   and    Unwin's   "Machine  De- 
sign," page  132. 


RIVETED   FASTENINGS  155 

square  inch.  It  is  shown  by  experiment  that  the  metal  between 
the  rivet  holes  has  a  higher  apparent  tensile  strength  than  that 
of  the  unperforated  plate;  this  increase  being  sometimes  as 
high  as  20%.  It  is  questionable,  however,  if  this  should  be 
taken  account  of  in  designing,  especially  where  the  holes  are 
punched,  as  the  operation  of  punching  may  more  than  offset 
this  peculiar  increase. 

52.3.  Factor  of  Safety.     In  boiler  work  the  factor  of  safety  is 
taken  at  about  5  which,  of  course,  brings  the  working  stress  well 
below  the  elastic  limit.     If  the  joint  is  to  be  subjected  to  hydraulic 
pressures,  where  heavy  shocks  may  have  to  be  withstood,  this 
factor  should  be  increased. 

52.4.  Practical  Rules.     It  has  already  been  noted  that  practi- 
cal considerations  make  it  'necessary  to  modify  the  theoretical 
equations  for  uniform  strength.     There  are  many  sets  of  practical 
rules  for  designing  riveted  joints  a  number  of  which  will  be 
found  in  the  references  given  below. 

Rules  of  the  Hartford  Steam  Boiler  and  Inspection  Co. 

Rules  of  the  American  Bureau  of  Shipping. 

Rules  of  the  Master  Steam  Boiler  Makers'  Association. 

Rules  of  the  U.  S.  Board  of  Supervising  Inspectors.  ' 

See  also  Cathcart's  "Machine  Design." 

Proceedings  of  Inst.  of  Mech.  Engineers,  1885. 

Unwin's  "Machine  Design." 

Wm.  M.  Barr's  "  Boilers  and  Furnaces." 

"  Steam  Boiler  Construction,"  W.  S.  Hutton. 


CHAPTER  VII 
SCREWS  AND  SCREW  FASTENINGS 

53.  Form  of  Screws.     Screws,  as  used  in  machines,  may  be 
divided  into  two  classes. 

(a)  Screw  fastenings. 

(b)  Screws  for  transmitting  power. 

The  form  of  the  thread  depends  upon  the  service  required. 
Thus,  for  screw  fastenings,  the  full  V  as  shown  in  Fig.  37  (a),  or 
modified  forms  of  V  threads,  as  shown  in  Fig.  37  (b)  and  (c),  are 
most  used  because  they  are  strong  and  easily  cut  by  machine 
dies.  They  are  inefficient  for  transmitting  power,  but'  this  is  a 
desirable  quality  in  fastenings,  as  it  reduces  the  liability  of  un- 
screwing. For  transmitting  power  the  square  thread,  Fig.  37  (d),  is 
most  used,  since  its  efficiency  is  higher  than  that  of  any  other 
form.  It  cannot  be  cut  with  a  die,  however,  and  it  is  difficult  to 
compensate  for  wear  with  this  form  of  thread.  For  these  reasons 
the  half  V  thread,  Fig.  37  (e),  is  often  used  for  transmitting  power 
when  wear  is  an  important  factor.  In  Fig.  37  (e)  the  Acme 
standard  thread  of  this  form  is  shown.  The  efficiency  of  this 
form  of  thread  is  a  little  less  than  the  square  thread  but  it  can  be 
cut  with  a  die  and  wear  can  be  compensated  for  by  means  of  a 
longitudinally  split  nut;  this  compensation  making  it  very  de- 
sirable for  such  .service  as  lead  screws  of  lathes,  etc.  Fig.  37  (f) 
illustrates  the  buttress  thread,  which  is  often  used  to  exert  pressure 
in  one  direction  only.  The  pressure  face  is  perpendicular  to 
the  axis  of  the  screw,  and  the  back  face  usually  makes  an  angle 
of  45°  with  this  axis.  This  screw  has,  therefore,  the  efficiency 
of  the  square  thread  and  the  strength  of  the  V  thread.  The 
underlying  principles  of  all  screws  are  the  same,  and  before  dis- 
cussing the  various  forms  and  classes  in  detail  the  fundamental 
equations  relating  to  their  action  will  be  developed. 

156 


SCREWS   AND    SCREW   FASTENINGS 


157 


54.  Friction  and  Efficiency  of  Square  Threaded  Screws.     In 

Fig.  38  (b),  let  N  represent  a  nut  moving  on  a  square  thread, 
under  the  action  of  a  tangential  force  P,  acting  at  the  mean 
radius  of  the  thread.  Let  this  force  P  be  applied  by  means  of  a 
couple,  so  that  there  is  no  lateral  pressure  against  the  screw. 
Let  W  represent  the  load  under  which  the  nut  is  moved,  and 
consider  that  it  can  move  only  in  an  axial  direction,  hence  there 
is  friction  between  N  and  W.  This  frictional  force  Fl  (Fig.  38  b) 
may  or  may  not  act  at  the  same  radius  as  P,  and  the  work  due 
to  this  frictional  force  will  vary  with  the  radius  at  which  it  acts. 
It  can  be  considered  as  forming  a  resisting  moment  opposing  the 


Full  V 
(a) 


Sellers  or  U.S.  Standard 
(b) 


Acme  Standard  Thread 
(«> 


Buttress  Thread 
(f) 


FIG.  37. 


turning  moment  due  to  P,  hence,  when  computing  the  re- 
quired turning  moment  of  P,  is  to  be  added  to  that  value.  It 
can  therefore,  for  simplicity,  be  omitted  temporarily  from  the 
discussion. 

Let  fi  =  coefficient  of  friction  between  thread  and  nut. 
Hi  =  coefficient  of  friction  between  load  and  collar. 

d  =  nominal  or  external  diameter  of  screw. 

r  =  nominal  or  external  radius  of  screw. 
d^  =  diameter  of  screw  at  bottom  of  thread. 
r   =  radius  of  screw  at  bottom  of  thread. 


158 


MACHINE    DESIGN 


r  =  frictional  radius  of  collar. 


mean  diam.  of  thread  = 


d+ 


rm  =  mean  radius  of  thread  = — • 

2 

s  =  pitch,  or  angular  advance  of  thread  per  turn. 
a  =  angle  made  by  thread  with  a  plane  perpendicular  to 
the  axis  of  screw. 


If  now  the  thread  be  developed  as  in  Fig.  38  (a),  it  is  seen, 
since  the  thread  is  a  true  helix,  that  the  action  of  the  thread 
and  nut  is  identical  with  that  of  a  body  N  sliding  up  an  inclined 
plane  of  length  *dm,  and  vertical  height  s  equal  to  the  pitch,  and 
carrying  a  load  W  which  is  free  to  move  vertically  only.  Omit- 
ting Fl  temporarily,  the  forces  acting  are  the  load  W,  the  fric- 
tion F2  between  the  thread  and  nut,  the  driving  force  P,  and  the 
normal  reaction  R.  It  is  required  to  determine  for  any  angle 
«,  the  value  of  P  required  to  slide  the  body  N  (turn  the  nut) 
up  the  incline.  The  frictional  resistance  F2  =  nR.  Hence, 
resolving  all  forces  parallel  to  ac 


SCREWS   AND   SCREW   FASTENINGS  159 

Pcos«  -  nR  =  Wsina      .     .      .     .     (i) 

P  COS  «  —  W  sin  a 
.-..*--  —  ....       (3) 

Resolving  all  forces  perpendicular  to  ac 


Substituting  in  (3)  the  value  of  R  obtained  in  (2 


p  .  W 

LCOS  a  —  fi  sin  a-l 

Since  (Fig.  38  a)  cos  a  =  -  —  and  sin  «  =  —  ,  equation  (4) 
may  be  written 


The  friction,  F^—Hi  PF,  and  if  rc  be  the  radius  at  which  Ft 
acts,  the  moment  of  Fl  around  the  axis  of  the  screw  =  fil  Wrc', 
and  when  this  resistance  is  considered  the  total  moment  of  P 
around  the  axis  is 


If  the  load  is  being  lowered,  the  directions  of  Fl  and  F2  are 
reversed,  and  in  this  case  the  turning  moment  that  must  be 
applied  is 


In  equation  6'  the  first  term  of  the  right-hand  side  of  the 
equation  is  the  moment  of  the  resistance  at  the  thread,  while 
the  second  term  is  the  moment  of  the  collar  friction.  If 

^  T:  dm  =  s,  that  is  if  — -  =  tan  a  =  /*,  the  moment  of  the  re- 

71  ^m 

sistance  at  the  thread  will  be  zero,  and  if  there  is  no  collar 
friction,  or  if  it  is  very  small,  as  in  the  case  of  ball-bearing 
thrusts,  this  will  give  a  condition  of  equilibrium,  the  friction  of 
the  thread  alone  just  sustaining  the  load,  and  Pl  will  be  equal  to 


l6o  MACHINE    DESIGN 

zero.  If  the  pitch  s  is  made  greater  than  p  *  dm,  the  moment  of 
the  resistance  at  the  thread  becomes  negative;  and  if  increased 
till  its  numerical  value  is  equal  to  the  moment  of  collar  friction, 
the  entire  right-hand  side  of  the  equation  will  be  equal  to  zero 
and  the  load  will  just  be  sustained  by  the  friction  of  the  thread 
and  collar  combined.  If  the  pitch  is  still  further  increased,  the 
entire  right-hand  side  of  the  equation  becomes  negative,  and 
the  moment  P±  rm  must  be  applied  in  the  direction  of  raising 
the  load,  or  the  screw  will  "overhaul,"  the  nut  exerting  a  turn- 
ing moment  in  the  downward  direction. 

To  find  the  limiting  value  of  «  where  the  nut  will  not  over- 
haul, equate  the  right-hand  side  of  the  equation  (6')  to  zero  and 

solve  for  -  — ,  whence 
*.dm 

s  v-  rm  +  PI  rc 

— -  =  tan  a  =  -  ....      (7) 

*dm  rm  —  plfirtt 

If  f  =  o,  in  which  case  there  is  no  moment  due  to  collar  fric- 
tion, tan  «  =  p  as  before. 

If  rm  —  rc  and  p  =  /^ 

tan  «  =  -37-,    ......      (8) 

and  if  p  be  taken  as  .1  (see  Art.  65)  tan  «  =  .2  whence 
a  =  n°. 

To  find  the  efficiency  of  the  screw,  consider  that  the  load  has 
been  raised  a  distance  equal  to  the  pitch,  that  /<,=/<,  and  rc  =  rm, 
then 

work  done  W  s  W  s 

e  =  efficiency  =  —  ,    ,  = —  =      ,    „  or 

energy  expended       2  ^  P  rm      T.  dm  P 

since  5  =  -  dm  tan  «,  and  inserting  the  value  of  P  from  equation  (5) 

W  TT  dm  tan  « 


am  —  p 

tan  «  (i  —  p  tan  «) 


nearly    ...      .     (o) 


tan  a  +  2  / 

If  the  collar  friction  is  zero,  or  very  small,  as  in  the  case  of 
ball-bearing  thrust  collars 


SCREWS   AND    SCREW   FASTENINGS  l6l 

tan  «  (i  —  ft  tan  «) 


tan  a 


(10) 


If,  in  equation  (9),  ft  be  taken  as  .1,  and  tan  «  as  .2  as  before, 
e  will  equal  50  per  cent  (nearly),  and  a  brief  reflection  will  show 
that  in  no  case  can  the  efficiency  of  a  self-sustaining  hoisting 
screw  exceed  50  per  cent.  Suppose  the  load  (Fig.  38)  to  be  just 
sustained  by  the  frictional  resistance  of  lowering,  that  is  tan  a 
«=  fj.  or  a  =  angle  of  repose.  If  now  a  force  P  is  ap- 
plied, just  sufficient  to  relieve  this  frictional  resistance,  the 
load  will  be  sustained  by  the  force  and  the  reaction  R.  If 
the  frictional  resistance  of  raising  were  zero,  the  slightest  ad- 
dition to  P  would  move  the  body  up  to  plane.  But  the  fric- 
tional resistance  of  raising  is  equal  to  that  of  lowering,  and  con- 
sequently, before  the  body  can  be  started  up  the  plane,  a  force 
2  P  must  be  applied;  which  is  twice  the  force  required  to  bal- 
ance the  frictional  resistance,  and  the  efficiency  would  then 
be  50  per  cent.  A  similar  reasoning  will  apply  to  other  hoisting 
devices  which  are  barely  self-sustaining  on  account  of  friction, 
namely,  that  the  force  which  must  be  applied  to  start  the  load  is 
equal  to  the  friction  due  to  lowering  plus  the  friction  due  to 
raising.  Hence  the  maximum  efficiency  for  such  self-sustaining 
mechanisms  is  50  per  cent. 

In  designing  screws  for  power  transmission  it  is  desirable  to 
know  the  pitch  angle  that  will  give  maximum  efficiency  for  the 
conditions  taken.  If  the  first  differential  of  equation  (9)  be 
taken  and  equated  to  zero,  it  is  found  that  the  maximum  efficiency 
when  collar  friction  is  considered  will  occur  when 


tan  «  =  V  2  +  4  /*2  —  2  /* 

If  /*  =  .  i  as  in  transmission  screws  where  lubrication  is  imper- 
fect, tan  «  =  1.23  and  «  =  51°. 

In  the  case  of  oil  bath  lubrication,  as  in  worm  gearing,  ft  may 
be  as  low  as  .05  when  tan  a  for  maximum  efficiency  =  1.318  or 
a  =  52°  -49'. 

In  a  similar  way  from  equation  (10)  for  maximum  efficiency 

tan  «  =  V  i  +  i?  —  fi 


1  62  MACHINE    DESIGN 

Whence  f  or  /*  =  .1,0  =  42° 
and  for//  =  .05,  «  =  43°~34' 

The  effect  of  the  pitch  angle  on  the  efficiency  of  the  screw  is 
of  great  importance  in  designing  screws  for  power  transmission, 
and  is  more  fully  discussed  in  Arts.  64  and  65. 

55.  Friction  and  Efficiency  of  Triangular  Threaded  Screws. 
With  triangular  threaded  screws  the  normal  pressure  at  the 
threads  is  greater  than  with  square  threads;  hence  the  friction 
at  the  threads  is  greater,  other  things  being  equal.  In  Fig.  38  (c) 
the  normal  pressure  for  a  square  thread  is  indicated  by  R,  while 
the  normal  pressure  for  a  triangular  thread  is  R'=R  sec  y,  in 
which  <p  =  half  the  angle  between  the  adjacent  faces  of  a  thread. 
R"  represents  the  radial  crushing  action  on  the  thread  of  the 
screw,  and  its  equal  and  opposite  reaction  tends  to  burst  the  nut. 
With  60°  angular  thread,  as  in  the  Sellers  system,  or  the  common 
V  thread,  R'  =R  sec  30°  =  i.i$R.  The  friction  increases  directly 
as  the  normal  pressure;  or  it  is  about  15  per  cent  greater  in  the 
60°  angular  thread  than  in  the  square  thread. 

If  in  equations  (i)  and  (3),^  sec  y  be  substituted  for  R,  then  by 
a  similar  method  of  reasoning  as  in  square  threads,  when  collar 
friction  is  neglected, 


P  .  w 


If  the  collar  friction  is  considered,  the  moments  around  the 
axis  when  raising  the  load  may  be  written 


The  efficiency  of  a  triangular  threaded  screw,  following  the 
same  reasoning  as  for  square  threaded  screws,  taking  rm  =  r., 
and  /*!=/<,  is 

Ws         W  ^tana  _ 
e"=2xrP~        -dP 


SCREWS    AND   SCREW    FASTENINGS 
W  TT  d  tan  « 


tan  « (i  —  ft  tan  «  sec  <p) 

or  e  =  — nearly 

tan  «  +  ^  sec  ?  +  p 


For  the  thread  on  a  one-inch  bolt  in  the  Sellers  system  tan  a 
—  .04,  and  taking  /*=.i,e  =  n%.  The  efficiency  of  the  threads 
on  standard  bolts  is  hence  seen  to  be  very  low  and  this,  as 


rrn 


FIG.  39. 

has  been  pointed  out,  is  an  advantage  in  fastenings  as  it  tends 
to  prevent  them  from  unscrewing. 

56.  Screw  Fastenings.  Screw  fastenings  are  used  for  hold- 
ing two  machine  parts  together  in  permanent  position,  or  for 
adjusting  one  part  relatively  to  another.  There  is  a  great  variety 
of  screw  fastenings  but  all  may  be  roughly  classified  as  follows: 
i.  Through-Bolts;  2.  Studs;  3.  Tap-Bolts  and  Cap- Screws; 
4.  Machine  Screws;  5.  Set  Screws. 

Through-Bolts.  A  through-bolt,  or  "bolt"  as  it  is  com- 
monly called,  Fig. 39  (a),  has  a  solid  head  on  one  end  and  a  nut  on 
the  other.  It  is  the  best  form  of  screw  fastening  and  should 


1 64  MACHINE    DESIGN 

always  be  used  when  the  hole  can  be  drilled  completely  through 
the  two  pieces  to  be  held  together. 

Studs.  Sometimes  it  is  not  possible  or  desirable  to  drill  a  hole 
entirely  through  both  pieces  which  are  to  be  held  together,  and 
in  such  cases  a  stud-bolt,  or  "stud,"  is  often  used.  A  stud, 
Fig.  39  (b),  is  a  circular  bar  having  a  thread  cut  on  each  end.  A 
hole  is  tapped  in  the  part  that  cannot  be  drilled  clear  through, 
and  one  end  of  the  stud  is  screwed  firmly  into  the  hole.  The 
part  that  can  be  drilled  through  is  slipped  over  the  stud,  and 
a  nut  on  the  outer  end  clamps  the  two  parts  firmly  together. 
Where  a  through-bolt  cannot  be  used  the  stud  is  the  next  best 
fastening.  It  should  be  a  tight  fit  in  the  tapped  hole,  and  when 
once  screwed  in  should  not  be  taken  out,  especially  if  the  hole  is 
tapped  into  cast  iron,  as  repeated  removal  wears  out  the  threads. 
The  length  of  the  tapped  hole  should  be  at  least  one  and  one  half 
times  the  diameter  of  the  stud,  in  order  to  secure  ample  frictional 
resistance  against  turning  when  the  nut  is  unscrewed. 

Tap-bolts  and  Cap-Screws.  Tap-bolts,  Fig.  39  (c),  and  cap- 
screws,  Fig.  39  (d),  have  a  solid  head  on  one  end  and  a  thread  on 
the  other.  They  are  used  under  exactly  the  same  circumstances 
as  the  stud  but  are  not  as  good  a  fastening,  as  they  necessarily 
must  be  unscrewed  from  the  tapped  hole  whenever  they  are  re- 
moved. Where  they  have  to  be  frequently  unscrewed,  and  espe- 
cially if  the  hole  is  tapped  into  cast  iron,  they  should  be  avoided. 
The  only  difference  between  tap-bolts  and  cap-screws  is  in  the  size 
and  form  of  the  head,  the  tap-bolt  having  a  standard  head  (see 
next  article) ,  and  the  cap-screw  for  the  same  size  of  bolt  having  a 
smaller  head  slightly  rounded  on  top.  Tap-bolts  are  much 
used  in  such  work  as  securing  patches  on  boilers,  where  a 
large  head  is  desirable.  Cap-screws  are  a  little  more  orna- 
mental and  are  much  used  in  cheaper  grades  of  machinery. 
They  are  a  standard  article  in  the  market  and  hence  can  be 
bought  very  cheaply. 

Machine  Screws.  Under  the  term  "machine  screws"  are 
included  many  forms  of  small  screws  usually  provided  with  a 
slotted  head  so  that  they  may  be  set  up  with  a  screw-driver.  The, 
most  usual  forms  of  machine  screws  are  shown  in  Fig.  (39  e,  f,  g 


SCREWS    AND   SCREW    FASTENINGS  165 

and  h) .  At  e  is  shown  an  oval  fillister  head ;  at  /  a  flat  fillister 
head ;  at  g  a  flat  countersunk  head ;  and  at  h  a  round  head. 

Machine  screws  are  designated,  for  convenience,  by  numbers, 
the  larger  numbers  indicating  the  larger  diameters.  Thus  the 
smallest  size,  as  given  in  Brown  and  Sharp's  catalogue,  is  number 
ooo  the  diameter  of  which  is  .03152.  The  difference  in  diameter 
between  consecutive  numbers  is  .01316.  The  diameter  of  a 
number  o  screw  is  .0578,  so  that  the  diameter  of  any  number 
larger  than  this  is  given  by  the  formula  d  =  .0131  N  +  .0578; 
where  d  is  the  diameter  in  inches,  and  N  the  serial  number  of 
the  screw.  The  number  N  is  not  to  be  confused  with  the  number 
of  threads  per  inch  n.  Machine  screws  larger  than  number  16, 
which  is  about  ]/±"  in  diameter,  are  not  much  used  in  machine 
work,  another  standard,  to  be  discussed  later,  being  used  for 
sizes  above  that  diameter. 

Manufacturers  have,  so  far,  been  unable  to  agree  upon  stand- 
ard numbers  of  threads  per  inch,  for  a  given  diameter  of  machine 
screw.  Thus  a  number  12  machine  screw  may  have  20  or  24 
threads  per  inch,  so  that  these  screws  are  usually  specified  by  nam- 
ing the  size  number  first,  followed  by  the  number  of  threads  per 
inch.  Thus,  an  18-20  machine  screw  means  size  18  and  20  threads 
per  inch.  Because  of  the  great  confusion  now  existing  regarding 
this  point,  the  American  Society  of  Mechanical  Engineers  ap- 
pointed a  committee  to  establish,  if  possible,  a  system  of  standards 
for  machine  screws.  This  committee  has  reported  and  their 
recommendations  can  be  found  in  Vol.  28  of  the  Transactions. 

Set  Screws.  Set  screws  are  a  form  of  screw-fastening  fre- 
quently used  to  prevent  relative  rotation  of  two  machine  parts. 
Thus  in  Fig.  40  the  hub  a  is  prevented  from  revolving  on  the 
shaft  b  by  the  set  screw  c.  The  head  of  the  set  screw  is  square 
while  the  point  may  be  cup-shaped  as  in  Fig.  40  (a),  round  as  in 
Fig.  40  (b) ,  or  conical  as  at  c  in  Fig.  40.  When  the  set  screw  is  made 
in  the  form  shown  at  Fig.  40  (a),  the  point  is  hardened  to  enable  it 
to  cut  into  the  shaft,  thus  increasing  its  holding  power.  If  the 
screw  is  made  of  tool  steel  the  hardening  may  be  done  by  the 
ordinary  process  of  tempering;  if  made  of  wrought  iron  the  same 
result  may  be  obtained  by  case  hardening.  The  objection  to 


1 66 


MACHINE    DESIGN 


the  cup-shaped  end  is  that  it  makes  a  burr  on  the  shaft  which 
sometimes  greatly  interferes  with  the  removal  of  the  hub.  To 
obviate  this  a  small  conical  depression  is  sometimes  made  in  the 
shaft  with  the  end  of  a  drill  and  the  form  shown  at  Fig.  40  used. 
Set  screws  are  not  reliable  for  heavy  work  and  should  be  used 
only  when  the  load  is  light. 

Standard  Screw  Threads.  Sellers  or  U.  S.  System.  Screw 
fastenings  larger  than  X  inch  diameter  are  made  according  to 
some  standard  system  in  order  to  secure  interchangeability. 
The  first  system  of  this  kind  was  that  introduced  into  England 


FIG.  40. 


by  Sir  Joseph  Whitworth.  The  form  of  the  Whitworth  thread 
is  shown  in  Fig.  37  (c).  The  thread  angle  is  55°  and  the  top  and 
bottom  of  thread  are  rounded  off  as  shown. 

The  recognized  standard  screw  thread  in  the  United  States  is 
the  Sellers,  U.  S.,  or  Franklin  Institute  thread.  The  form  of 
this  thread  is  shown  in  Fig.  39  (b).  The  thread  angle  is  60°; 
the  top  of  the  thread  is  cut  off  and  the  bottom  of  the  thread 
filled  in  as  shown.  This  standard  is  not  used  exclusively  in 
this  country,  however,  but  a  full  V  thread,  as  shown  in  Fig.  37  (a), 
without  the  flattened  tops  and  bottoms  is  also  in  common  use. 
The  angle  of  such  V  thread  is  generally  60°  in  machine  bolts 
and  the  number  of  threads  per  inch  usually  corresponds  to  those 
of  the  Sellers  system,  but  there  are  many  variations  in  this  par- 
ticular. Where  the  Sellers  standard  is  not  strictly  adhered  to 


SCREWS    AND    SCREW    FASTENINGS 


i67 


it  is  advisable,  therefore,  to  buy  machine  bolts  of  one  manufac- 
turer only  or  so  specify  as  to  insure  interchangeability. 

The  Sellers  screws  have  much  greater  tensile  strength  than 
screws  with  full  V  threads  of  equal  angles  and  pitch,  because  the 
thread  of  the  former  is  only  three-fourths  as  deep  owing  to  the 
flattening  at  the  tops  and  bottoms. 

TABLE  X 

SELLERS,    U.    S.,    OR     FRANKLIN    INSTITUTE    STANDARD    BOLTS 


The  area  of  a  i"  full  60°  thread  is  .482  square  inches,  while 
the  area  at  the  bottom  of  a  Sellers  thread  is  .55  square  inches, 
or  14  per  cent  greater. 

The  Whitworth  system  of  threads  differs  from  the  Sellers  in 
the  shape  of  the  thread,  as  noted  above,  and  the  number  of 
threads  per  inch  is  also  different  for  some  diameters.  Thus  the 


l68  MACHINE    DESIGN 

Sellers  system  gives  13  threads  per  inch  for  %"  bolts  while  the 
Whitworth  gives  12.  The  Whitworth  system  gives  somewhat 
stronger  screws  as  the  diameter  at  the  root  of  the  thread  is 
greater  for  the  same  size  of  bolt  and  the  rounded  shape  at  the 
root  is  stronger  than  the  flat  root  of  the  Sellers  thread.  The 
Sellers  thread  is,  however,  much  easier  to  produce  than  the 
Whitworth. 

Table  X  gives  the  proportion  of  screws  as  fixed  by  the  Sel- 
lers standard  for  bolts  up  to  2^".  Above  this  size  the  stand- 
ard is  not  adhered  to  rigidly,' as  the  size  and  pitch  of  the  screw 
becomes  rather  large  for  convenience.  Thus  a  6"  bolt  in  the 
Sellers  system  will  have  2%  threads  per  inch.  It  is  common, 
therefore,  to  make  these  larger  sizes,  which  are  comparatively 
rare,  with  4  threads  to  the  inch. 

In  Germany,  France,  and  other  European  countries  other  sys- 
tems are  in  use,  some  of  which  are  based  on  metric  units. 

57.  Pipe  Threads.     The  Briggs    system  of    pipe  threads  is 
the  established  standard  in  the  United  States.     The  numbers  of 
threads  per  inch  for  the  various  sizes  of  pipe  are  given  below : 

yi"  pipe,  27  threads  per  inch. 
yi"  and  y%"  pipe,  18  threads  per  inch. 
y2"  and  fan  pipe,  14  threads  per  inch, 
i"  to  2"  pipe,  nK  threads  per  inch. 
2%ff  and  over,  8  threads  per  inch. 

For  form  of  threads  and  other  details  as  to  Briggs  system, 
see  Trans.  A.  S.  M.  E.,  Vol.  VIII,  page  29. 

58.  Straining  Action  in  Bolts  due  to  External  Load.     The 
load  applied  to  the  bolts  is  generally  one  which  tends  to  separate 
the  connected  members,  in  the  direction  of  the  axis  of  the  bolt,  and 
this  action  is  resisted  by  a  tensile  stress  in  the  bolts;  but  bolts  are 
sometimes  used  to  prevent  the  relative  translation  of  two  or  more 
pieces,  when  a  shearing  stress  is  produced  in  the  bolts.     When 
the  action  of  the  load  is  oblique  to  the  axis,  the  stress  in  the  bolt 
may  be  combined  tension  and  shearing. 

If  any  screw  is  tightened  up  under  load  there  is  an  initial 
direct  stress  (tension  or  compression)  and  usually  a  torsional  stress 


SCREWS   AND    SCREW   FASTENINGS  169 

due  to  friction  between  the  threads  of  the  screw  and  the  nut. 
With  bolts  or  studs  screwed  up  hard,  as  in  making  a  steam- 
tight  joint,  the  initial  tension  due  to  screwing  up  may  be  much 
in  excess  of  that  due  to  the  working  load.  This  will  be  treated 
more  fully  later. 

If  the  load  applied  to  the  bolt  produces  a  shearing  action,  the 
bolt  shank  should  accurately  fit  the  holes  in  the  connected  pieces, 
at  least  for  the  portions  near  the  joint  ;  and  if  P  is  the  load  per  bolt, 
d  the  diameter  of  the  bolt  (shank),  and  pa  the  shearing  stress, 


In  a  bolt  subjected  to  a  load  which  produces  tension,  the  mini- 
mum cross  section  sustains  the  greatest  stress.  This  smallest 
cross  section,  in  common  bolts,  is  through  the  bottoms  of  the 
threads.  Thus  if  a  load  P  be  applied  to  an  eye-bolt,  as  in  Fig. 
41,  the  only  stress  that  will  be  induced  in  the  bolt  will  be  that 
due  to  the  external  load  P.  If  p  be  the  tensile  stress  due  to  the 
load  P,  and  di  the  diameter  of  the  bolt  at  the  bottom  or  root  of 
the  threads, 


Values  of  d^  are  given  in  Table  X,  page  167,  for  the  various 
sizes  of  Sellers  screws.  For  a  given  diameter  and  pitch  of  screw 
the  area  at  the  bottom  of  threads  would  be  considerably  less  with 
full  V  threads. 

59.  Initial  Tension  in  Bolts  due  to  Screwing  up.  If  the 
bolt  is  used  simply  to  hold  two  machine  parts  together,  as  in 
Fig.  39  (a),  and  there  is  no  external  load  tending  to  separate  the 
parts,  the  stress  in  the  bolt  will  be  the  resultant  of  the  tensile 
stress  due  to  screwing  up  the  nut,  and  the  torsional  stress  due 
to  the  frictional  resistance  at  the  thread. 

In  the  Sellers  system  the  pitch  angle  of  the  thread  (a)  varies 
from  about  3°  in  a  %"  screw,  to  i°-5o'  in  a  3"  screw;  or  tan  « 
varies  from  .054  to  .032  in  this  same  range.  If,  therefore,  in 
equation  (12),  rc  be  taken  equal  to  |  rm,  n  be  taken  at  .1,  and  ^ 
.15,  it  appears  that  P  varies  from  .35  W  with  a  K"  screw  to  .32 


1 70  MACHINE    DESIGN 

W  with  a  3"  screw.  The  coefficients  of  friction  will  vary  much 
more  than  this,  so  it  may  be  assumed  that  for  the  ordinary  range 
of  screw  fastenings 

P  =  -33  W  approximately 

p 

or  the  tension  in  the  screw  W  =  — . 

•33 

The  turning  moment,  Prm,  due  to  the  wrench  pull,  is  resisted 
by  the  frictional  moment  of  the  nut  or  collar  and  the  frictional 
moment  at  the  thread.  This  frictional  moment  at  the  thread  is 
transmitted  to  the  body  of  the  bolt,  so  that  the  bolt  itself  is  sub- 
jected to  a  twisting  moment  equal  to  Prm  minus  the  frictional 
moment  at  the  nut  or  collar.  The  resultant  stress,  therefore, 
under  these  circumstances,  is  that  due  to  combined  twisting  and 
direct  stress,  and  it  can  be  shown  (see  Art.  67)  that  the  resultant 
stress  as  determined  Jby  equation  (i),  page  49,  is  from  15  to  20 
per  cent  greater  than  the  direct  stress  alone. 

Refined  calculations  regarding  the  resultant  stress  in  bolts  due 
to  screwing  up  are,  in  general,  useless  and  misleading,  especially 
in  the  case  of  fastenings  less  than  ]/%"  in  diameter.  Since  a  me- 
chanic using  a  wrench  of  ordinary  proportions  can  easily  rupture 
any  of  these  small  fastenings,  it  follows  that  the  actual  stress  in 
such  bolts  depends  entirely  on  the  judgment  of  the  mechanic. 

A  series  of  experiments  was  made  in  the  Sibley  College  Labo- 
ratory, a  few  years  ago,  to  determine  directly  the  probable  load 
produced  in  standard  bolts  when  making  a  tight  joint.  The  sizes 
of  bolts  used  were  %" ,  y^ ',  i"  and  1%".  One  set  of  experiments 
was  made  with  rough  nuts  and  washers,  and  another  set  with  the 
nuts  and  their  seats  on  the  washers  jfaced  off.  A  bolt  was  placed 
in  a  testing  machine,  so  that  the  |fcdal  force  upon  it  could  be 
weighed  after  it  was  screwed  up.  Each  of  twelve  experienced 
mechanics  was  asked  to  select  his  own  wrench  and  then  to  screw 
up  the  nut  as  if  making  a  steam-tight  joint,  and  the  resulting  load 
on  the  bolt  was  weighed.  Each  man  repeated  the  test  three  times 
for  every  size  of  bolt,  and  each  had  a  helper  on  the  i"  and  i%n 
sizes,  The  sizes  of  wrenches  used  were  10"  or  1 2"  on  the  Y*"  bolts 
up  to  1 8"  and  22"  on  the  i%"  bolts.  The  results  were  rather  dis- 


SCREWS    AND   SCREW    FASTENINGS  171 

cordant,  as  should  be  expected;  the  loads  in  the  different  tests 
were  rather  more  uniform,  as  well  as  higher,  with  the  faced  nuts 
and  washers.     The  general  results  indicate:    (a)  that  the  initial 
load  due  to  screwing  up  for  a  tight  joint  varies  about  as  the 
diameter  of  the  bolt;    that  is,  a  mehanic  will  graduate  the  pull 
on  the  wrench  in  about  that  ratio,     (b)  That  the  load  produced 
may  be  estimated  at  16,000  Ibs.  per  inch  of  diameter  of  bolt,  or 

W  =  16,000  d  ......      (15) 

in  which  W  is  the  initial  load  in  pounds  due  to  screwing  up,  and 
d  is  the  nominal  (outside)  diameter  of  the  screw  thread.  This 
value  of  W  is  rather  above  the  average  for  the  tests;  but  it  is 
considerably  below  the  maximum,  and  it  is  probably  not  in  ex- 
cess of  the  load  which  may  reasonably  be  expected  in  making  a 
tight  joint. 

If  the  initial  load  due  to  screwing  up  be  divided  by  the  Cross- 
sectional  area  of  the  bolt  at  the  bottom  of  the  threads,  the  initial 
intensity  of  the  tensile  stress  is  obtained.  The  above  experi- 
ments indicate  that  this  intensity  of  stress  varies,  approximately, 
inversely  as  the  nominal  diameter  (d)  of  the  bolt;  and  that  it 
may  frequently  equal  or  exceed 

P  =  3°'°00  Ibs.  per  sq.  in.    .  (16) 

In  addition  to  this  tensile  stress  there  is,  as  before  stated,  a 
considerable  twisting  action  on  the  bolt.  Equation^  (i  6)  would 
give  a  stress  of  60,000  Ibs.  per  square  inch  on  a  K-inch  bolt; 
and  this  result  is  substantiated  by  the  fact  that  steel  bolts  of  this 
size  were  broken  in  the  course  of  the  experiments.  It  also  agrees 
with  common  experience  which  forbids  the  use  of  screws  as  small 
as  K-inch  for  cases  re^uiri^g  .the  nuts  to  be  screwed  up-  hard. 

In  these  experiments,  the  average  effective  lever  arm  of  the 
wrench  was  not  far  from  15  times  the  diameter,  or  30  times  the 
radius,  of  the  screw;  hence,  if  it  be  assumed  as  in  the  previous 
paragraph  that  the  turning  force  acting  at  the  radius  of  the  screw 
is  P  =  .^W  the  force  applied  at  the  wrench  is,  in  pounds,  about 


P    ..-  -          _  -33  X  16,000  <*  ^ 

3°          3°  3° 


172  MACHINE    DESIGN 

The  above  discussion  indicates  that  the  factor  of  safety  should 
be  increased  as  the  size  of  the  screw  decreases,  and  of  course  this 
factor  should  be  varied  with  the  conditions  of  the  case,  as  in 
some  applications  the  nuts  are  much  more  apt  to  be  screwed  up 
hard  than  in  others.  *. 

A  set  of  experiments  was  made  by  Mr.  James  McBride  (Trans. 
A.  S.  M.  E.,  Vol.  XII,  page  781),  which  show  that  the  factor  of 
safety,  as  bolts  are  frequently  used,  is  very  low,  even  with  a  very 
moderate  external  load.  One  case  cited  by  Mr.  McBride  indi- 
cates that  the  stress  due  to  -screwing  up  a  3^ -inch  bolt  was 
nearly  one-half  the  ultimate  strength,  or  probably  very  near  the 
elastic  limit.  His  direct  determinations  of  the  efficiency  of  a 
standard  2-inch  screw  bolt  shows  an  average  of  only  10.19  per 
cent.  It  is  probably  this  low  efficiency  which  saves  many  screws 
from  being  broken,  as  the  frictional  loss  reduces  the  tension  pro- 
duced in  the  bolt  by  screwing  up.  The  excessive  friction  makes 
the  screw  bolt  a  useful  fastening,  as  it  reduces  the  tendency  to 
"overhaul"  or  unscrew. 

60.  Resultant  Stress  on  Bolts  due  to  Combined  Initial  Tension 
and  External  Load.  It  was  shown,  in  Art.  59,  that  bolts 
may  be  subjected  to  a  high  tensile  stress  by  screwing  up  the 
nuts.  The  question  often  arises  as  to  the  effect  of  the  combined 
action  of  this  initial  tension  and  the  external,  or  useful,  load. 
It  is  stated  by  some  that  the  resultant  load  on  the  bolt  is  simply 
the  sum  of  the  initial  and  the  external  loads.  Others  contend 
that  the  application  of  the  external  load  does  not  change  the 
stress  in  the  bolt,  unless  this  external  load  exceeds  the  initial  load 
due  to  screwing  up;  that  is,  that  the  resultant  load  is  equal  to 
the  initial  load  alone,  or  to  the  external  load  alone,  whichever  is 
the  greater. 

Neither  of  these  views  is  entirely  correct  for  conditions  at- 
tained in  practice.  They  represent  the  extreme  limiting  cases 
and  every  actual  case  lies  between  them. 

If  the  bolt  itself  could  be  absolutely  rigid  while  the  members 
forced  together  in  screwing  it  up  yielded  under  pressure,  the  total 
load  on  the  bolt  would  be  equal  to  the  sum  of  the  initial  load  and 
the  external  load.  If,  however,  the  members  pressed  together 


SCREWS   AND    SCREW    FASTENINGS 


were  absolutely  rigid,  only  the  bolt  yielding,  the  total  (resultant) 
load  on  the  bolt  would  be  the  initial  load  alone,  or  the  external 
load  alone,  whichever  is  the  greater. 

The  first  of  the  above  conditions  is  approached  by  the  arrange- 
ment shown  in  Fig.  42.  Screwing  up  the  nut  compresses  the 
spring  interposed  between  A  and  B.  Assume  that  an  axial  force 
of  2,000  pounds  will  compress  this  spring  i  inch;  then  if  the  nut 
is  screwed  up  till  the  spring  is  2  inches  shorter  than  its  free  length, 
the  load  on  the  bolt,  due  to  screwing  up,  must  equal  the  reaction 


FIG.  46. 


FIG.  48. 


FIG.  44. 


FIG.  42. 


FIG.  43. 


FIG.  45. 


of  the  spring,  or  4,000  Ibs.  Assume,  also,  that  the  extension  of 
the  bolt  under  this  screwing-up  action,  or  under  the  initial  load 
of  4,000  Ibs.,  is  .02  inch.  Now,  if  an  external  axial  load  of  say 
2,000  Ibs.  be  applied  to  the  eye  at  the  bottom  of  B,  this  added  load 
would  tend  further  to  increase  the  length  of  the  bolt  by  about  .01 
inch;  but  this  further  extension  of  the  bolt  would  reduce  the 
compression  on  the  spring  by  a  corresponding  amount  and  thus 
slightly  diminish  the  spring  reaction.  With  such  great  differ- 
ence between  the  rigidity  of  the  bolt  and  of  the  connected  mem- 
bers, the  load  on  the  bolt  becomes  practically  the  sum  of  the 


174  MACHINE    DESIGN 

initial  and  the  external  loads,  but  the  resultant  load  is  necessarily 
somewhat  less  than  this  sum  in  any  possible  case. 

The  arrangement  shown  in  Fig.  43  is  one  which  approaches 
the  other  limiting  case  mentioned  above.  Suppose  the  bolt  to  be 
a  spring  which  is  subjected  to  an  axial  load  of  4,000  Ibs.  in  screw- 
ing the  nut  up  two  inches,  and  that  the  corresponding  yielding  of 
the  member  B  is  .02  inch.  The  initial  load  on  the  bolt  (which  is 
the  spring  in  this  case)  is  4,000  Ibs.,  and  the  pressure  between 
the  contact  surfaces  of  A  and  B  is  equal  to  it.  If  an  external 
axial  load  be  now  applied  to  the  eye  in  B,  the  pressure  between 
the  contact  surfaces  is  reduced  by  an  amount  nearly  equal  to  this 
external  load.  But,  unless  the  external  load  exceeds  the  initial 
load,  the  bolt  will  not  elongate  enough  to  separate  these  contact 
surfaces  and  entirely  remove  the  pressure  between  them,  because 
the  load  on  the  bolt  (spring)  cannot  change  without  changing  the 
length  of  the  bolt,  and  with  the  above  data  the  bolt  would  have  to 
stretch  an  additional  .02  inch  (equal  to  the  initial  yielding  of  the 
connected  members)  before  the  contact  surfaces  would  be  entirely 
relieved  of  pressure.  It  therefore  appears  that  the  addition  of  an 
external  load  in  this  case  does  not  materially  affect  the  resultant 
tension  on  the  bolt  as  long  as  this  external  load  does  not  exceed 
the  initial  load.  If  the  external  load  is  greater  than  the  initial 
load  (say  6,000  Ibs.),  the  elongation  of  the  bolt  increases  (to  3 
inches) ;  the  resultant  load  on  the  bolt  will  be  simply  the  external 
load  alone,  because  the  latter  is  sufficient  entirely  to  relieve  the 
pressure  produced  between  the  contact  surfaces  in  screwing  up. 

In  all  ordinary  practical  cases  the  difference  in  rigidity  between 
the  bolt  and  the  connected  members  is  much  less  than  in  the  ex- 
treme conditions  considered  above.  The  resultant  load  on  a  bolt 
may  be  anything  between  the  sum  of  the  initial  and  the  external 
loads  as  a  maximum,  and  the  greater  of  these  two  loads  alone  as  a 
minimum.  This  resultant  load  approaches  the  maximum  limit 
when  the  bolts  are  rigid  relative  to  the  connected  members  as  in 
Fig.  44;  and  this  resultant  approaches  the  minimum  limit  when 
the  bolts  are  relatively  yielding,  as  in  Fig.  45.  In  any  particular 
case  the  designer  can  tell  which  limit  is  the  more  nearly  ap- 
proached, and  he  should  be  governed  accordingly. 


SCREWS   AND    SCREW   FASTENINGS  175 

The  Locomotive  (Nov.,  1897)  contains  an  excellent  article 
on  the  resultant  load  on  bolts,  and  a  relation  is  derived  from 
which  the  following  method  of  treatment  has  been  developed: 
The  application  of  this  method  depends  simply  upon  the  ratio  of 
the  yield  of  the  connected  members  to  the  yield  of  the  bolts.  It 
will  usually  not  be  difficult  to  assign  a  sufficiently  close  value  to 
this  ratio  even  when  the  actual  magnitudes  of  yielding  are  un- 
known; in  fact,  only  a  rough  approximation  to  the  value  of  this 

y 

ratio  is  necessary.     Let  this  ratio  be  called  y  and  let— - —  =  x: 

y  +  i 

call  the  initial  load  on  the  bolt  due  to  screwing  up  W^  the  exter- 
nal (useful)  load  TF2;  and  the  total  (resultant)  load  W.^  Then  it 
can  be  shown  that 

W  =  Wl  +  x  W2. 

If  the  yield  ratio  (y)  is  known,  the  value  of  x  is  at  once  found  by 
the  above  relation  of  oc  and  y.  If  the  yield  of  the  connected  mem- 
bers is  between  i  and  5  times  that  of  the  bolt,  the  resultant  load 
is  equal  to  the  initial  load  added  to  from  0.5  to  0.8,  the  external 
load.  If  a  tight  joint  is  made  with  short  rigid  bolts  or  studs,  con- 
necting flanges  which  are  separated  by  an  elastic  packing,  or  with 
a  metal  contact  at  some  distance  from  the  center  line  of  the  bolts, 
as  indicated  in  Fig.  44,  the  applied  load  is  an  important  consider- 
ation since  the  value  of  y  is  relatively  great.  In  some  other  cases 
the  external  load  may  be  a  minor  consideration  as  affecting  the 
strength  of  the  bolt. 

When  the  conditions  are  such  that  the  nut  is  not  apt  to  be 
screwed  up  hard,  that  is  when  the  initial  load  may  be  safely 
neglected,  design  for  the  external  load  alone. 

The  following  suggestions  may  serve  as  a  guide  in  practical 
problems  involving  the  resultant  load  on  bolts  when  the  initial 
load  due  to  screwing  up  is  apt  to  be  considerable. 

(a)  If  a  bolt  is  manifestly  very  much  more  yielding  than  the 
connected  members,  design  the  bolt  simply  for  the  initial  load  or 
for  the  external  load,  whichever  is  the  greater. 

(b)  If  the  probable  yield  of  the  bolt  is  from  one-half  to  once 
that  of   the  connected    members,   consider  the  resultant   load 


176  MACHINE    DESIGN 

as  the  initial  load  plus  from  one-fourth  to  one-half  the  external 
load. 

(c)  If  the  yield  of  the  connected  members  is  probably  four  or 
five  times  that  of  the  bolts,  take  the  resultant  load  as  the  initial 
load  plus  about  three-fourths  the  external  load. 

(d)  In  case  of  extreme  relative  yielding  of  the  connected  mem- 
bers, the  resultant  load  may  be  assumed  at  nearly  the  sum  of  the 
initial  and  external  loads. 

61.  Allowable  Stress  in  Screw  Fastenings.  From  the  fore- 
going it  is  seen  that  small  screw  fastenings  are  very  liable  to  be 
heavily  overstrained  by  the  initial  load  due  to  screwing  up  the 
nut.  While  the  body  of  the  bolt  is  well  designed  to  resist  heavy 
loads  a  source  of  weakness  is  found  in  the  threaded  portion. 
The  reduced  area,  due  to  cutting  the  thread,  localizes  the  greatest 
stress,  and  cracks  are  very  liable  to  start  from  the  roots  of  the 
threads,  especially  where  the  thread  is  of  the  full  V  form. 

For  these  reasons  the  ordinary  apparent  fibre  stresses  allowed 
in  most  machine  members  cannot  be  permitted  in  screw  fasten- 
ings. For  ordinary  purposes,  where  overstraining  is  not  likely 
to  occur,  or  for  large  bolts,  8,000  to  10,000  Ibs.  per  square  inch 
may  be  allowed,  for  steel.  For  such  work  as  steam  and  hydraulic 
joints,  where  the  initial  stress  may  be  large,  from  6,000  to  8,000 
Ibs.  per  square  inch  should  be  allowed,  depending  on  the  condi- 
tions and  quality  of  material  employed,  and  if  shocks  are  liable 
to  occur,  stresses  as  low  as  3,000  to  4,000  are  often  preferable, 

Example  i :  The  cylinder  of  the  steam  engine  is  12  inches  in 
diameter,  and  the  cylinder  head  is  held  in  place  by  10  steel 
through  bolts.  The  maximum  steam  pressure  is  100  pounds  per 
square  inch.  If  the  contact  surfaces  of  the  head  and  cylinder 
are  ground  together  so  that  no  packing  is  necessary,  what  must 
be  the  diameter  of  the  bolts  so  that  the  maximum  stress  in  the  bolt 
necessary  to  insure  a  steam-tight  joint  will  not  exceed  7,000  Ibs. 
per  square  inch  ? 

In  this  case  it  is  evident  that  the  bolts  are  much  more  yielding 
than  the  parts  which  they  hold  together  and  the  conditions  are 
those  of  case  a  in  the  previous  paragraph.  It  is  also  clear  that 
the  initial  load  on  the  bolt  must  be  greater  than  the  external 


SCREWS   AND   SCREW   FASTENINGS  177 

load  due  to  the  steam  pressure  in  order  to  insure  a  steam-tight 
joint.  If  this  initial  load  be  taken  at  twice  the  external  load  a 
fair  margin  of  safety  is  secured.  If  W^  be  the  initial  load  and 
W2  the  external  load  per  bolt  then 


4X  10 

whence  W{  =  2  W2  =  2,260  Ibs. 
Whence  if  d1  be  the  diameter  at  the  root  of  the  thread 

^ 

-  X  7,000  =  2,260. 


Therefore  dl  =  .64  inch  (at  root  of  thread) 

which  corresponds  closely  to  a  K7'  screw.  It  is  to  be  noted  that 
while  a  total  stress  of  7,000  Ibs.  per  square  inch  of  section  is 
sufficient  to  insure  a  tight  joint,  a  much  greater  stress  may  be  in- 
duced by  the  workman  in  screwing  up  the  nut,  if  he  is  careless 
or  inexperienced. 

Example  2  :  If  in  the  above  example  steel  studs  are  used  and 
rubber  packing  y%"  thick  be  placed  between  the  contact  surfaces, 
what  must  be  the  diameter  of  the  studs  ? 

Here  the  parts  held  together  are  more  elastic  than  the  studs 
and  the  conditions  may  be  taken  as  corresponding  to  those  of 
case  c.  As  before,  the  initial  load  W1  may  be  taken  at  twice  the 
external  load. 

TT  X  i22 
Then  W2  =  -  —  X  100  =  1,130 

and  Wt  =  2  W2  =  2,260. 
From  (c)  paragraph  60  the  total  load 

W  =  W,  +  X  W2 
=  2,260  +  (X  X  1,130)  =  3,107  Ibs. 

*  d* 

Whence  -    -  X  7,000  =  3,107 
4 

and  dl  =  .76  inch  (at  root  of  thread) 
which  corresponds  closely  to  a  %  inch  screw. 

12 


178  MACHINE    DESIGN 

The  maximum  stress  which  the  workman  may,  perhaps,  in- 
duce in  the  stud  by  screwing  up  the  nut  is 

30.000       30,000 

p  =  -  — —  =  -  — —  =  34,000  IDS.  approximately, 
a  y% 

which  will  be  increased  a  little  by  the  external  load.  This  is 
close  to  the  elastic  limit;  but  it  may  be  noted  that  even  should 
the  elastic  limit  be  slightly  exceeded  the  efficiency  of  the  fast- 
ening is  not  impaired,  since  here  permanency  of  form  is  not  so 
essential  as  in  machine  parts  which  transmit  motion. 

62.  Resilience  of  Bolts  with  Impulsive  Load. 

In  bridge  work  and  other  cases  requiring  long  bolts,  it  is  very 
common  to  make  the  cross-section  through  the  body  of  the  bolt 
about  equal  to  the  section  at  the  bottom  of  the  threads.  This 
may  be  done  by  upsetting  the  ends  where  the  thread  is  to  be  cut, 
or  by  welding  on  ends  made  from  stock  somewhat  larger  than 
that  used  for  the  main  length  of  the  bolt. 

The  most  apparent  result  of  this  practice  is  to  economize  mate- 
rial without  sacrifice  of  strength  (as  the  shank  still  has  an  area  of 
cross-section  equal  to  the  threaded  portion),  and  if  the  weld 
(when  the  ends  are  welded)  is  perfect,  the  strength  of  the  bolt 
is  not  reduced.  It  seems  probable  that  this  reason  is  responsible 
for  the  original  adoption  of  this  practice,  since  it  has  been  most 
generally  used  in  long  tie  rods.  However,  in  case  of  bolts  liable 
to  shock,  there  is  an  even  more  important  reason  for  such  con- 
struction ;  since  it  can  be  shown  that  the  reduced  section  not  only 
maintains  the  full  strength  under  static  load,  but  it  very  greatly 
increases  the  capacity  of  the  bolt  to  resist  shock.  This  last  fact 
has  not  been  very  generally  recognized,  as  appears  from  the  com- 
mon application  of  such  reduced  shank  bolts  only  to  structures, 
rather  than  to  machines. 

It  has  been  seen  that  the  resistance  of  a  tension  member  under 
a  static  load  is  determined  solely  by  its  weakest  section;  while, 
in  a  member  subjected  to  shock,  impact,  or  impulsive  load,  the 
resistance  depends  upon  the  total  extent  of  distortion  of  the 
member  due  to  a  given  intensity  of  stress. 

As  shown  in  Art.  24,  the  maximum  stress  with  impulsive  load  is 


SCREWS   AND   SCREW    FASTENINGS  179 

W(h  +  S) 
kd  A 

For  a  stress  within  the  elastic  limit 
2W     h  + 


This  shows  clearly  that  for  a  given  load,  W,  applied  suddenly 
or  with  impact,  the  stress  produced  in  a  member  of  sectional  area, 
A,  is  greater  as  d  becomes  less  relative  to  h.  Hence,  if  d  is  in- 
creased, the  stress  produced  becomes  less  for  a  given  impulsive 
action;  or  the  resistance  to  such  action  is  greater  for  a  given 
value  of  the  maximum  stress. 

If  an  ordinary  bolt  is  subjected  to  shock  in  a  direction  to  pro- 
duce tension,  the  stress  will  be  a  maximum  at  the  sections  through 
the  bottom  of  the  threads;  the  bolt  will  elongate,  but  the  elonga- 
tion will  be  confined  largely  to  the  very  short  reduced  (threaded) 
sections,  hence  the  stress  will  be  much  less  in  the  larger  portion 
of  the  bolt.  In  a  Sellers  '  bolt  of  one  inch  diameter  the  area  A  of 
the  shank  is  .78  sq.  inches,  while  the  area  A1  at  the  bottom  of 
threads  is  only  .55  sq.  inches.  Therefore  a  stress  on  A'  of  30,000 

Ibs.  per  sq.  in.  =  —    —  -  —  —  =  21,000  on  the  full  sections.     Sup- 

pose the  elongation  per  inch  of  length  at  a  stress  of  30,000  (taken 
as  the  elastic  limit)  is  roVo".  Each  inch  of  section  A'  will  elongate 
ToW>  while  each  inch  of  full  section  A  (  =  .78  sq.  in.)  will  have 
a  stress  of  only  21,000  Ibs.,  with  a  corresponding  elongation  of 
M  X  ToVo  =  -0007".  Assume  the  thread  to  be  i"  long,  and  the 
remainder  of  the  bolt  to  be  5"  long.  It  will  appear  that  the  mean 
stress  on  the  threaded  portion  (i")  is  about  the  mean  of  30,000 
and  21,000,  or  say  25,500  Ibs.  per  square  inch;  as  the  mean  sec- 
tion is  an  average  of  .55  and  .78  square  inches.  Hence  the 
elongation  for  this  threaded  i  inch,  when  the  stress  on  4'  =  30,- 
ooo,  is  .00085",  while  the  other  5"  (of  area  A)  will  elongate  under 
this  load  5  X  .0007  =  .003  5".  The  total  elongation  will  then  be 
3  =  .00085  +  -°°35  =  -°°435  inches. 


l8o  MACHINE    DESIGN 


*  ,  x     ,  8,50  x  ,04l6  ,  344  lbs. 


Now,  suppose  the  5"  shank  of  this  bolt  were  reduced  in  section 
to  an  area  ^'  =  .55.  Then  the  elongation  of  this  portion  under 
the  above  load  would  be  5  X  .001  =  .005",  instead  of  .0035"  and 
the  total  elongation  would  be  d  =  .00085  +  .005  =  .00585. 


.<<  X  30,000       . 
-W-->     -*-i     -  X  .--  =  8250  X  0.553  -  457 


This  latter  load  is  33  per  cent  greater  than  the  preceding. 

The  preceding  example  shows  that  the  elastic  resilience  of  the 
bolt  was  increased  33  per  cent  by  reducing  the  body  of  the  bolt 
to  A'.  Of  course  the  gain  would  be  still  greater  with  a  longer 
bolt.  It  may  be  well  to  remember  that  the  "long  specimen"  is 
more  apt  to  contain  a  weak  section  than  is  a  short  specimen;  but, 
on  the  other  hand,  the  sharp  notching  of  the  threads  is  quite 
liable  to  start  a  fracture  at  their  roots. 

If  the  bolt  is  strained  beyond  the  elastic  limit,  the  portion  thus 
strained  yields  at  a  much  greater  rate,  relative  to  the  stress,  than 
that  given  above.  With  a  load  which  would  produce  a  stress  of 
30,000  lbs.  per  sq.  in.  in  the  larger  portion  (area  A),  the  stress  in 

the  reduced  portion  (area  A')  will  be  —  —  -  =  43,000  lbs. 

*0  0 

per  sq.  inch.  Hence,  the  effect  of  a  long  section  in  resisting 
shock  without  rupture  is  much  greater  even  than  that  shown  for 
elastic  deformation  only. 

The  section  of  the  shank  of  the  bolt  may  be  reduced  as  in  Fig. 
46,  by  turning  down  the  body  of  the  bolt  to  about  the  diameter 
at  the  bottoms  of  the  threads.  The  collars  a  and  a'  may  be  left 
to  form  a  fit  in  the  hole.  This  form  is  easy  to  make,  but  does 
not  fit  the  hole  throughout  its  length,  and  it  is  weak  in  torsion. 

Fig.  47  is  somewhat  more  expensive,  but  fits  the  hole  better, 
and  is  somewhat  stronger  in  torsion.  Fig.  48  is  the  form  which 
gives  the  best  fit,  and  is  also  the  strongest  in  torsion.  If  very 
long  it  is  difficult  to  make;  otherwise  it  is  perhaps  the  best. 


SCREWS   AND    SCREW   FASTENINGS  l8l 

These  high  resilience  bolts  only  increase  the  resistance  to  im- 
pulsive load,  not  to  dead  load.  They  are  good  forms  to  use  in 
such  cases  as  the  so-called  " marine  type"  of  connecting  rod, 
where  the  bolts  are  subjected  to  considerable  shock. 

For  cylinder  head  bolts,  and  other  cases  where  a  tight  joint  is 
the  main  consideration,  this  form  of  bolt  may  be  entirely  unsuited. 

Professor  Sweet  prepared,  for  tests,  some  bolts  such  as  are  used 
in  the  connecting  rod  of  the  Straight  Line  Engine;  of  these,  half 
were  solid  (ordinary  form)  bolts,  and  the  other  half  were  of  the 
form  shown  in  Fig.  48. 

Tests  of  a  pair  of  these  bolts,  one  of  each  kind,  showed  an  elon- 
gation at  rupture  of  .25"  for  the  solid  bolt,  which  broke  in  the 
thread;  while  the  drilled  bolt  elongated  2.25",  or  9  times  as 
much,  and  it  broke  through  the  shank,  the  net  section  of  which 
was  a  trifle  less  than  that  at  the  bottom  of  the  threads.  Drop 
tests  showed  similar  results.  These  tests  indicate  the  superior 
ultimate  resilience  of  the  reduced  shank  bolts. 

It  was  shown  in  Art.  24,  page  77,  that  where  a  machine 
member  must  absorb  considerable  shock,  a  rather  weak  yielding 
material  might  be  safer  than  one  which  is  stronger  and  stiffer, 
because  of  the  greater  elastic  resilience  of  the  weaker  and  more 
ductile  material.  This  principle  is  of  importance  in  designing 
fastenings  which  are  subjected  to  shock  where  they  must  neces- 
sarily work  under  high  stress. 

63.  Location  of  Fastenings.  As  previously  stated,  screw 
fastenings  are  generally  intended  to  be  tension  members  only, 
and  from  the  foregoing  discussion  it  appears  that  even  when  used 
in  this  manner  alone  they  are  subjected  to  very  high  stresses. 
The  conditions  under  which  a  fastening  is  to  be  used  should 
therefore  be  carefully  considered  in  order  that  all  forces  acting 
upon  it  may  be  provided  for.  Further,  the  location  of  the 
fastening  may  or  may  not  be  advantageous,  thus  greatly  affect- 
ing its  required  size.  Thus  in  Fig.  49,  if  the  bolts  alone  are  de- 
pended upon  to  resist  the  downward  force  P,  they  must  be  care- 
fully fitted,  to  insure  that  each  bolt  receives  its  full  share  of  this 
shearing  load.  Through  bolts  only  can  be  used  in  such  a  case 
as  studs  or  tap  bolts  cannot  be  accurately  fitted.  If  the  down- 


182 


MACHINE    DESIGN 


ward  force  is  resisted  by  a  projecting  ledge,  as  at  A,  which  "is 
the  preferable  way,  the  bolts  need  not  fit  the  holes  closely  and 
either  studs  or  tap  bolts  can  be  used.  The  bracket  now  tends 
to  rotate  around  A  and  the  moment  of  the  load  P  I  must  equal 
the  sum  of  the  moments  of  the  bolts  round  the  same  point.  It 
is  evident  that  the  lower  bolt  must  be  considerably  larger  than 
the  upper  bolt,  to  be  equally  effective.  In  small  work  it  is  con- 
venient to  make  all  bolts  the  same  size,  the  sum  of  their  resist- 
ing moments  being  made  equal  to  P  I.  In  large  work  the  bolt 
at  C  is  often  made  large  enough  to  exert  a  moment  equal  to  P  /, 
and  the  bolts  near  A  serve  only  to  insure  correct  location.  The 
upper  bolt  should,  in  any  case,  be  located  as  far  away  from  A  as 
possible. 


In  many  machine  parts,  such  as  flywheel  rims  and  brake  bands, 
it  often  occurs  that  the  bolts  cannot  be  placed  directly  in  line 
with  the  applied  force  but  must  be  at  a  distance  /  (Fig.  50)  from 
its  line  of  action.  The  bolts  in  such  cases  may  be  subjected  to 
both  flexure  and  direct  stress.  Thus  in  Fig.  50,  if  the  bolts  fit 
the  holes  in  the  lugs  tightly  such  a  combination  of  stresses  will 
be  induced.  In  such  parts  as  brake  bands  the  connecting  bolts 
are  often  used  as  a  means  of  adjustment  against  wear,  as  shown 
in  Fig.  51.  If  in  such  a  case  the  lugs  be  weak  and  yielding,  the 
threaded  portion  of  the  bolt  will  be  subjected  to  both  flexure  and 
direct  stress.  The  threaded  portion  of  the  bolt  is  particularly 
weak  against  flexural  stress  because  cracks  are  easily  started  at 
the  root  of  the  threads,  and  where  screws  are  used  in  this  manner 
they  should  be  designed  with  a  large  factor  of  safety. 


SCREWS    AND   SCREW    FASTENINGS  183 

64.  Screws  for  the  Transmission  of  Power.  It  has  been 
pointed  out,  in  Art.  53,  that  the  square  thread  is  most  used  for 
transmitting  power  because  of  its  higher  efficiency,  and  that 
when  wear  must  be  compensated  for  the  half  V  thread  is  most 
serviceable  (see  Figs.  37  d  and  e  ).  Where  the  thread  angle  of  the 
half  V  thread  is  small,  as  in  the  Acme  thread,  the  general  equa- 
tions which  have  been  deduced  for  the  square  thread  may  be 
used  without  great  error. 

Equation  (6),  Art.  54,  expresses  the  relation  which  exists  be- 
tween the  turning  moment  which  must  be  applied  to  the  screw, 
and  the  moments  due  to  the  load,  friction  at  the  thread  and  at  the 
thrust  collar.  An  examination  of  this  equation  shows  that  for  a 
given  applied  force  P,  the  load  W  which  can  be  overcome  de- 
creases with  an  increase  in  the  value  of  the  pitch  s,  and  in- 
creases with  a  decrease  in  the  value  of  s,  since  5  is  added  to  the 
numerator  of  the  fraction  and  subtracted  from  the  denominator. 
This  can  be  seen  in  another  way  by  considering  the  energy  sup- 
plied and  the  work  performed.  If  the  force  P  be  applied 
through  a  complete  revolution,  or  a  distance  of  *  dm,  the  load 
will  be  raised  a  distance  equal  to  the  pitch  s.  Evidently,  if  s  is 
decreased,  a  greater  load  can  be  raised  by  a  given  force  P;  since 
(neglecting  friction),  the  force  applied,  multiplied  by  the  space 
through  which  it  moves,  must  be  equal  to  the  load  multiplied  by 
the  space  through  which  it  is  raised.  In  other  words,  the  me- 
chanical advantage  of  the  screw  can  be  varied  by  reducing  the 
pitch  angle;  and  it  is  evident  that  by  reducing  the  pitch  angle  a 
small  force  applied  at  a  long  radius  may  be  made  to  raise  a  great 
load. 

In  order  that  the  thread  on  the  screw  and  nut  may  be  equally 
strong,  with  similar  materials,  the  thread  and  space  on  the  screw 
are  made  equal  to  each  other  and  therefore  equal  to  half  the 
pitch.  As  the  pitch  is  increased  the  axial  width  of  both  thread 
and  space  are  necessarily  increased,  and  if  it  is  desired  to  keep 
the  section  of  the  thread  square  in  form,  this  soon  results  in  a 
very  heavy  thread  when  compared  to  the  cylinder  on  which  it  is 
formed.  If  the  depth  of  the  space  is  reduced,  to  avoid  reducing 
the  diameter  of  the  cylinder,  the  bearing  surface  of  the  screw 


184  MACHINE    DESIGN 

and  nut  is  reduced,  which  is  not  desirable.  It  is  customary  in 
such  cases  to  divide  the  axial  width  of  the  thread  and  space  into 
several  equal  parts,  arranged  alternately  round  the  axis  of  the 
screw,  thus  forming  several  parallel  threads  and  spaces.  The 
depth  of  the  space  can,  by  this  means,  be  greatly  reduced  and 
ample  wearing  surface  be  provided.  Such  screws  are  called 
multiple  threaded  screws  and  may  have  two,  three,  or  more 
parallel  threads.  The  theory  of  such  screws  is  evidently  iden- 
tical with  that  of  the  single  threaded  screw. 

65.  Friction  arid  Efficiency  of  Screws  for  Power  Trans- 
mission. Equations  (9)  and  (10),  while  expressing  the  general  re- 
lations which  exist  between  efficiency  and  the  pitch  angle,  do  not 
show  clearly  the  effect  upon  the  efficiency  due  to  varying  this 
angle.  In  Fig.  52  these  equations  have  been  plotted  for  various 
constant  values  of  /*,  and  these  curves  show  graphically  the  effect 
of  varying  the  pitch  angle.  An  examination  of  this  figure  shows 
that  for  the  value  of  p  chosen  the  efficiency  increases  rapidly,  as 
the  angle  increases,  up  to  15°  or  20°,  attaining  a  maximum  be- 
tween 40°  and  50°,  and  then  decreasing  with  an  increase  in  the 
angle,  becoming  zero  again  near  90°. 

It  is  to  be  noted  that  between  20°  and  60°  the  efficiency  does 
not  vary  materially  with  change  of  angle,  and  that  when  the 
efficiency  of  the  screw  alone  is  considered,  steep  pitched  threads, 
as  from  30°  to  50°  pitch  angle,  give  maximum  efficiency  and 
hence  a  more  durable  thread.  It  is  seldom  feasible  to  use  such 
pitches  in  practice,  for  reasons  that  will  be  presently  discussed. 
The  curves  in  Fig.  52  will  be  found  useful  in  making  trial 
assumptions  for  the  efficiency  of  screws. 

Screws  for  transmitting  power  are  usually  difficult  to  lubricate 
freely,  hence,  in  general,  their  rubbing  surfaces  are  imperfectly 
lubricated  (see  Art.  28).  The  coefficient  of  friction  for  screws 
working  under  pressures  ranging  from  3,000  to  10,000  Ibs.  per 
square  inch,  and  at  low  velocities,  has  been  experimentally  de- 
termined *  by  Professor  Albert  Kingsbury.  From  his  experi- 
ments it  appears  that,  for  these  conditions,  the  value  of  /*  may  be 

*  Transactions  of  American  Society  of  Mechanical  Engineers,  Vol.  17. 


£  Efficiency 


O»    *•     CO    1C    H> 


I  i  1  1  1 


1=  -ft     M  Ts    g 

II     II  II 


1 86  MACHINE    DESIGN 

taken  at  .15.  For  pressures  lower  than  3,000  Ibs.  per  square 
inch,  and  velocities  above  50  ft.  per  minute,  the  value  of  //  may 
be  assumed  at  .1,  if  fair  lubrication  is  maintained. 

The  bearing  pressure  per  unit  area  or  thread  surface  that 
may  be  carried  on  a  screw  thread,  will  vary  greatly  with  the 
conditions  of  service.  If  the  velocity  is  low,  and  wear  not  an 
important  factor,  as  in  the  case  of  jack  screws,  very  heavy  pres- 
sures may  be  carried;  but  where  accuracy  of  form  is  important, 
and  where  the  velocity  exceeds  50  ft.  per  minute,  the  pressure 
per  unit  area  should  not  exceed  200  Ibs.,  and  for  such  service  as 
lead  screws,  where  maintenance  of  form  is  essential,  it  should  be 
as  low  as  possible. 
If  W  =  load  carried. 

p  =  intensity  of  pressure  per  unit  of  projected  area  of  thread. 
n  =  number  of  threads  per  inch. 
/  =  length  of  nut  in  inches. 
d  =  outside  diameter  of  thread. 
Ji=  inside  diameter  of  thread. 

Then  W  =  pnl-^—d*]  ....      .     (17) 
4 

The  load  per  unit  of  projected  area  is  the  same  as  the  load  per 
unit  of  true  area,  since  the  projected  area  is  equal  to  the  true 
area  multiplied  by  cos  «,  and  the  axial  or  projected  pressure  is 
equal  to  the  normal  or  true  pressure  multiplied  by  the  same 
function. 

66.  Stresses  in  Transmission  Screws.  It  has  been  shown 
in  Art.  59  that  the  resisting  moment  at  the  thread  is,  from 

[~s  +  ft  -  d  n 

equation  (6),  equal  to  W  rm  I   —-—  or  to  the  total  turning 

*  tfm      s  M 

moment  applied,  minus  the  frictional  moment  at  the  collar. 
This  moment  induces  a  torsional  stress*  in  the  screw.  The 
direct  action  of  the  load  is  to  induce  a  tensile  or  compressive 

*  This  statement  applies  strictly  to  the  most  usual  case  only,  where  the  thrust 
collar  is  located  at  that  end  of  the  screw  to  which  the  power  is  applied.  If  the 
collar  is  not  located  at  the  end  to  which  the  power  is  applied,  the  total  torque, 
Prm,  of  equation  (6)  is  transmitted  through  the  body  of  the  screw. 


SCREWS   AND   SCREW    FASTENINGS  187 

W 
stress  in  the  screw  equal  to  —  (where    A    is  the  area  at  the 

A 

root  of  the  thread)  if  the  screw  is  short.  If  the  screw  is  over 
six  times  as  long  as  its  least  diameter  d1}  the  compressive  stress, 
if  the  screw  is  in  compression,  will  be  that  due  to  W,  considering 
the  screw  as  a  long  column.  Equations  (i)  and  (2)  of  Chapter 
III,  page  48,  and  their  discussion  in  Art.  18,  are  therefore,  ap- 
plicable to  the  design  of  such  screws. 

67.  Design  of  Screws  for  Power  Transmission.  An  in- 
spection of  Fig.  52  shows  that  screws  of  small  pitch  have  very 
low  efficiency,  and  it  would  seem  desirable  for  that  reason  to 
keep  the  pitch  as  great  as  possible.  On  the  other  hand,  it  was 
pointed  out  in  Art.  64  that  the  mechanical  advantage  of  a  screw 
increases  as  the  pitch  decreases.  It  was  also  shown  in  Art.  54 
that  a  self-sustaining  screw  could  not  have  an  efficiency  of  over 
50  per  cent,  and  for  perfect  safety  against  overhauling  it  should 
be  much  less  than  this  value.  The  best  pitch,  for  a  given  set  of 
conditions,  may  therefore  be  a  compromise  between  these  con- 
flicting requirements.  Thus  if  the  turning  moment  which  can  be 
applied  to  a  screw  is  limited  (as  is  often  the  case  with  hand 
power)  a  low  pitch  must  be  selected  in  order  to  attain  mechanical 
advantage.  In  such  a  case  it  is  obvious  that  care  should  be  used 
in  selecting  the  materials  of  the  screw  and  nut,  so  as  to  obtain  as 
low  a  coefficient  of  friction  as  possible.  Thus  a  bronze  nut  will 
run  well  on  a  steel  screw  with  imperfect  lubrication.  (See  Art. 
28.)  Again  in  such  cases  as  the  screws  in  certain  machine 
tools,  as  plate  planers,  a  more  efficient  pitch  may  be  taken.  If 
there  is  no  tendency  for  the  screw  to  overhaul,  and  the  necessary 
moment  can  be  applied,  the  pitch  of  maximum  efficiency  can  be 
selected.* 

Example.  The  force  required  to  open  or  close  a  certain  sub- 
merged sliding  water  gate  is  estimated  at  6,000  Ibs.  It  is  re- 
quired to  design  a  single-threaded  steel  screw  such  that  one  man 

*  In  certain  forms  of  saw-mill  carriages  these  conditions  exist,  and  the  screws 
for  setting  the  log  over  to  the  saw  may  be  and  are  made  with  a  very  efficient 
pitch. 


1  88  MACHINE    DESIGN 

exerting  a  pull  of  60  Ibs.  at  the  periphery  of  a  hand  wheel  40 
inches  in  diameter,  attached  directly  to  the  screw,  can  operate 
the  gate.  The  greatest  unsupported  length  of  the  screw  is 
found  to  be  4  ft.  Let  the  coefficient  of  friction  =  .15,  the  crush- 
ing strength  of  the  material  =  30,000  Ibs.  per  square  inch,  the 
maximum  working  tensile  or  compressive  stress  =  10,000  Ibs.  per 
square  inch,  and  the  coefficient  of  elasticity  =  3  0,000,000. 

Since  the  screw  is  in  compression  in  closing  the  gate  it  must 
be  designed  as  a  long  column  square  at  both  ends  (case  4),  and 
if  so  designed  it  will  have  surplus  strength  when  in  tension. 
The  effect  of  the  thread  in  stiffening  the  screw  is  small  and  will 
be  neglected.  The  total  maximum  unit  stress  in  the  screw  is  to 
be  10,000  Ibs.  and  it  is  evident  that  the  maximum  compressive 
stress  p  will  form  the  larger  part  of  the  total  stress;  p  may  there- 
fore be  taken  at  9,000  Ibs.  per  square  inch,  and  the  mean  com- 
pressive stress  p'  assumed  at  6,000  Ibs.  per  square  inch  :  whence 
the  trial  area  of  the  screw  at  the  root  of  the  thread 

=  T/"  =  7       -  =  i  sq.  inch,  or  a  diameter  of  i}i  inches.     Check- 
p        0,000 

ing  this  assumption  by  formula  N  page  94 


*"ZV     ^77V=  l^T    ^^  =  5,3°olbs. 


which  checks  closely  enough  with  the  mean  stress  assumed. 

Assume  the  efficiency  of  the  screw  and  collar  at  1  5  per  cent. 
The  energy  which  the  operator  can  supply  in  one  complete  revo- 
lution of  the  wheel  =  ^X  40  X  60  =  7,600  in.  Ibs.  Hence  the 
energy  delivered  at  the  nut  will  be  7,600  X  .15  =  1,140  inch  Ibs. 
But  during  one  revolution  of  the  screw  the  gate  must  move  a  dis- 
tance equal  to  the  pitch  s,  against  a  force  of  6,000  Ibs. 

Hence  s  X  6,000  =  1,140 

.'.  s  =  -r  -  =  .19"  or  say  -J-"  =  .2"  so  that  the  thread  may 

be  easily  cut  in  a  lathe. 

Since  the  thread  is  square  the  outer  diameter  of  the  screw  will 


SCREWS   AND   SCREW   FASTENINGS  189 

be  d^  +  5  =  iy&"  +  .2  =  1.325,  or  in  order  to  use  a  standard  tap 
the  outer  diameter  may  be  taken  as  i  X".  The  corrected  diame- 
ter of  the  screw  at  the  root  of  the  thread  will  be  1.5  — .2  =  1.3* 
and  the  corrected  mean  diameter  will  be  1.5—  .1  =  1.4".  For 

.2 

these    values    tan    «  =  —        -  =  .046  which    corresponds    to 

~    X     1-4 

«  =  2°  —  40'.  From  curve  5  (Fig.  52)  the  efficiency  of  the  screw 
is  about  13  per  cent,  and  the  original  assumption  of  efficiency 
is  sufficiently  close. 

The  twisting  moment  applied  to  the  screw  =  60X20  =  1,200 
in.  Ibs.  The  frictional  moment  at  the  collar  is  approximately 
/«  W  rm  =  .15  X  6,000  X  .7  =  630  in.  Ibs.  Hence  the  torsional 
moment  at  the  nut  * 

T  =  60  X  20  —  630  =  570  in.  Ibs. 

The  torsional  stress  due  to  this  moment  is  by  equation  E, 
page  91, 

i6T        16  X  570 

P  =  --TT  = '         No  =  1,320  Ibs. 

ndi3       *  x  (1-3) 

.'.by  equation  (I),  page  (48),  the  maximum  direct  stress, 
pm  =  Y*  [p  +  Vp2  +  4  ps2]  =  1A  [9,000  +  \/9,ooo2+ (4X1,320)'] 
=  9,600  Ibs.,  which  is  less  than  the  assigned  limit  and  the 
design  is  therefore  correct. 

The  increase  in  the  maximum  stress  due  to  the  torsional  mo- 
ment is  here  only  about  6  per  cent.  Where  the  screw  is  short,  so 
that  a  much  greater  mean  direct  stress  can  be  carried,  or  where 
the  screw  is  only  in  tension  and  hence  admits  of  a  high  tensile 
stress,  this  increase  may  be  from  15  to  20  per  cent.  If  the  screw 
is  made  of  cast  iron  it  should  also  be  checked  for  shearing  by 
equation  (2)  of  Art.  16. 

*  It  is  assumed  that  the  collar  is  at  the  upper  end  where  the  power  is  applied. 


CHAPTER  VIII 
KEYS,  COTTERS,  AND  FORCE  FITS 

68.  Forms  of  Keys.  Keys  are  wedge-shaped  pieces,  generally 
made  of  steel,  which  are  used  primarily  to  prevent  relative  rota- 
tion between  shafts  and  the  pulleys,  gears,  etc.,  which  they  carry. 
On  account  of  the  frictional  resistance  which  they  induce  between 
the  surface  of  the  shaft  and  the  member  which  is  keyed  to  it,  they 
also  often  prevent  relative  sliding  of  the  parts.  Keys  are  most 
usually  rectangular  in  cross-section;  but  occasionally  they  are 
made  of  circular  form.  A  saddle  key  is  shown  in  Fig.  53. 
This  form  of  key  does  not  require  the  shaft  to  be  cut;  but  its 
holding  power  is  so  small  that  it  is  used  only  for  light  work.  For 
small  loads,  or  as  a  safeguard  when  the  hub  is  shrunk  on,  a  round 
pin  as  shown  in  Fig.  54  (a)  is  often  used.  Figs.  54  (b)  and  54  (c) 
show  two  other  methods  of  applying  round  taper  pins  as  a  sub- 
stitute for  keys.  A  flat  key  is  shown  in  Fig.  55.  This  form 
requires  a  small  portion  of  the  shaft  to  be  cut  away,  and  its  hold- 
ing power  is  much  greater  than  that  of  the  saddle  key.  The  sunk 
key,  Fig.  56,  is  the  most  secure  form  of  key  fastening,  and  is  more 
used  than  any  other.  It  is  so  called  because  it  is  sunk  into  a 
keyway  or  groove  cut  in  the  shaft.  It  thus  requires  more  metal 
to  be  cut  away  from  the  shaft  than  the  flat  key,  and  this  must  be 
taken  into  account  in  designing  shafting  since  the  metal  is  removed 
from  the  outer  fibre  where  it  is  most  serviceable  for  resisting 
applied  loads.  The  keyway  cut  in  a  shaft  for  a  sunk  key  is  made 
parallel  to  the  axis  of  the  shaft;  but  the  keyway  in  the  hub  of  the 
pulley  or  gear  which  is  to  be  made  fast  is  cut  tapering  as  shown 
in  Fig.  56  (b).  The  sides  of  the  key  are  parallel,  as  shown,  and 
should  fit  well  in  both  shaft  and  hub.  When  the  key  is  driven 

190 


KEYS,    COTTERS,    AND    FORCE    FITS 


TQI 


in,  the  shaft  and  hub  are  drawn  tightly  together  on  the  side  of 
the  shaft  opposite  to  the  key,  and  the  frictional  resistance  thus 
set  up  helps  to  prevent  relative  sliding  of  the  parts  lengthwise 
of  the  shaft.  If  the  bore  of  the  hub  is  tapering,  or  if  the  key  fits 


FIG.  53.  FIG.  54  (a).        FIG.  54  (b).        FIG.  54  (c).         FIG.  55. 

more  tightly  at  one  end  than  at  the  other,  the  part  keyed  on  may 
be  thrown  out  of  alignment  so  that  its  plane  is  not  perpendicular 
to  the  axis  of  the  shaft.  Where  great  accuracy  is  required,  as  in 
flanged  couplings  on  shafting,  owing  to  this  tendency,  the  faces 
of  the  flange  or  part  secured  to  the  shaft  should  be  faced  in  place 
after  the  key  is  driven.  If  the  part  keyed  on  does  not  have  to 
be  removed  often,  the.  hub  may  be  made  a  tight  or  press  fit  on 
the  shaft,  thereby  preventing  largely  the  tilting  action  of  the  key 
should  such  occur.  In  the  Woodruff  system  (Fig.  57),  the  key 


FIG.  57. 


is  a  circular  segment  and  the  keyway  may  be  cut  with  a  milling 
cutter.  This  allows  the  key  to  adjust  itself  to  the  taper  of  the 
keyway  in  the  hub,  hence  it  will  not  throw  the  keyed  part  out 
of  perpendicular  alignment.  With  this  system,  the  hub  must 


MACHINE    DESIGN 

be  forced  on  over  the  key.  These  keys  are  used  largely  in  ma- 
chine tools. 

In  general,  the  part  to  be  secured  on  the  shaft  is  placed  in 
position  and  the  key  driven  in.  This  makes  it  necessary  to  ex- 
tend the  keyway  along  the  shaft  at  least  the  length  of  the  key, 
(except  when  the  hub  is  at  the  end  of  the  shaft)  unless 
the  diameter  of  the  shaft  is  enlarged  under  the  hub,  suffi- 
ciently to  allow  the  keyway  to  be  cut  without  cutting  into  the 
shaft  proper.  Where  it  is  desirable  to  withdraw  the  key  occasion- 
ally, it  is  often  provided  with  a  head,  as  shown  in  Fig.  56  (b),  in 
which  case  it  is  called  a  draw  key.*  Sometimes,  however,  it  is 
not  desirable  to  extend  the  keyway  beyond  the  hub,  in  which 
case  the  keyway  in  the  shaft  is  made  the  same  length  as  the  key, 
and  the  hub  is  driven  over  the  key  into  its  correct  position. 
Much  more  force  is  necessary  to  drive  the  hub  into  place  in  this 
manner  than  to  drive  the  key,  on  account  of  the  friction  between 
the  shaft  and  the  hub.  When  the  hub  is  a  sliding  or  an  easy  fit 
on  the  shaft,  and  only  one  key  is  used,  there  is  a  tendency  to 
throw  the  hub  eccentric  to  the  shaft.  Under  these  circumstances 
there  is  a  tendency  for  the  hub  to  rock  and  work  loose  on  the 
shaft,  especially  if  the  direction  of  motion  be  reversed.  In  such 
cases  two  keys  set  90°  apart  make  a  much  more  secure  fastening 
as  this  gives  three  lines  of  contact  and  prevents  rocking.  If  one 
of  these  keys  is  a  saddle  key,  as  shown  in  Fig.  53,  the  fitting  is 
greatly  facilitated  and  the  fastening  is  almost  as  secure  as  with 
two  sunk  keys. 

69.  Stresses  in  Sunk  Keys.  Since  keys  are  designed  to  prevent 
relative  rotation,  it  is  evident  that  every  key  must  transmit  a 
certain  torsional  moment  or  torque.  This  torsional  moment 
may  be  equal  to  the  total  torque  transmitted  by  the  shaft,  or  the 
key  may  be  required  to  transmit  only  a  part  of  it.  This  would 
indicate  that  keys  of  different  sizes  should  be  used  with  any  given 
diameter  of  shaft,  depending  on  the  load  which  the  key  must 
transmit.  For  practical  reasons,  however,  such  as  standardiza- 

*  Where  a  draw  key  cannot  be  used  the  point  of  the  key  is  sometimes  case- 
hardened  so  that  it  will  not  upset  so  readily  in  being  driven  out 


KEYS,    COTTERS,    AND    FORCE    FITS  193 

tion  and  interchangeability,  it  is  desirable  that  the  dimensions 
of  the  shaft  and  key  should  bear  a  fixed  relation  to  each  other. 
All  practical  systems  of  keys,  therefore,  give  a  fixed  size  of  key 
for  each  diameter  of  shaft,  the  dimensions  of  the  key,  presuma- 
bly, being  such  that  its  strength  is  equal  to  the  torsional  strength 
of  the  shaft.  Shafts  are  usually  designed  for  torsional  stiffness 
rather  than  torsional  strength,  which  results  in  a  shaft  consider- 
ably larger  than  necessary  as  far  as  strength  is  concerned.  If, 
under  these  circumstances,  the  key  is  designed  as  indicated  above 
it  will  also  have  excess  strength.  Where  the  shaft  is  short  and 
is  designed  for  strength  alone,  the  key  should  be  more  carefully 
considered. 

Keys  resisting  a  torsional  moment  are  subjected  to  simple 
crushing,  or  to  crushing  and  shearing,  depending  on  the  man- 
ner of  their  application  and  manner  of  fitting.  The  ordinary 
sunk  key  (Fig.  56  a),  is  subjected  to  a  force,  Fly  due  to  the 
pressure  from  the  shaft,  and  to  a  resisting  force,  F2,  due  to 
the  reaction  from  the  hub  which  it  secures.  The  effect  of 
these  two  forces  is  to  set  up  a  shearing  stress  along  the  middle 
section  of  the  key  at  the  outer  surface  of  the  shaft.  They  also  form 
a  couple  which  tends  to  rotate  the  key  in  the  keyway.  This  tend- 
ency to  rotate  should  be,  for  best  results,  resisted  by  the  pressure 
of  the  hub  and  shaft  against  the  top  and  bottom  of  the  key.  If 
the  key  is  not  a  tight  fit  on  the  top  and  bottom  these  resisting 
pressures,  F3  and  F4,  will  be  concentrated  near  the  corners.  This 
concentrated  pressure  may  be  sufficient  to  crush  the  key  at  these 
points,  and  allow  it  to  roll  in  the  keyway,  deforming  both  the 
keyway  and  key  and  subjecting  the  key  to  a  severe  crushing 
action  rather  than  simple  shear.  If  the  conditions  of  service 
require  a  continual  reversal  of  motion  a  state  similar  to  that 
shown  in  Fig.  56  (c)  is  induced,  where  the  resisting  forces  F3 
and  F±  have  been  moved  inward  and  their  moment  arm  made  so 
short  that  their  magnitude  must  be  very  great  to  hold  the  key  in 
position.  This  may  bring  a  severe  bursting  stress  on  the  hub. 
It  is  evident,  therefore,  that  keys  which  fit  sidewise  only,  cannot 
be  depended  on  to  carry  as  great  a  load  as  those  which  fit  well 
on  the  top  and  bottom.  Where  great  accuracy  is  required,  as 
13 


IQ4  MACHINE    DESIGN 

in  machine  tool  construction,  the  hub  is  often  made  a  force  fit 
on  the  shaft  and  the  key  fitted  only  on  the  sides,  so  that  it  cannot 
throw  the  parts  out  of  relative  alignment  by  radial  pressure. 
Referring  to  Fig.  56  (a) 
Let    /  =  the  length  of  the  key  or  hub 
"    /=    "  thickness  of  the  key 
"    b=    "  breadth  of  the  key 
«    f=   u  torsional  moment  applied  to  the  shaft 
"    P=  "  force  acting  at  the  radius  of  the  shaft  so  that 

P-  =  T 

2          .  ... 

Then  for  shearing  stress  pa 

P-P.l* (i) 

and  since  the  torsional  moment  applied  to  the  shaft  must  equal 
the  moment  of  the  crushing  load  applied  to  the  side  of  the  key 

T  =  Pl  =  pia 

d  t  fd       t 

2         c    2  \  2       4 
If  Fv  be  considered  to  act  at  the  radius  of  the  shaft  (which  can 
be  done  without  serious  error  for  keys  as  ordinarily  proportioned) 
equation  (2)  reduces  to 

P-PJ^ •••     (3) 

Equations  (i)  and  (3)  may  be  used  to  compute  the  stresses  in  any 
sunk  key. 

If  the  shearing  resistance  of  the  key  is  to  equal  the  crushing 
resistance,  then  from  (i)  and  (3) 

p  I  b  =  p  I  — 

rs  rc     2 


If  pc  =  2  pa,  t  =  b,  and  the  key  is  square  for  equal  resistance  to 
shearing  and  crushing.     For  machinery  steel,  such  as  is  generally 

A 

used  in  keys,  —  =  .8,  and  hence  from  (4)  for  equal  strength  in 


KEYS,    COTTERS,    AND    FORCE    FITS  IQ5 

shearing  and  compression  t  =  i.6b.  If,  in  addition,  the  moment 
of  the  shearing  resistance  of  the  key  is  to  be  equal  to  the  torsional 
resisting  moment  of  the  shaft,  then 

d  *d? 

T-P.lb--f.-x    .....     (5) 

where  p'a  is  the  shearing  stress  in  the  outer  fibre  of  the  shaft. 
For  steel  shafts  and  keys,  which  are  most  common,  pa  =  p's  whence 
from  (5) 

d       *  d3 
lb-  =  -   -  .      .      .....     (6) 

2        16 

The  minimum  length  of  hub  (/)  ,  as  determined  by  practice,  which 
is  necessary  to  give  a  good  grip  on  the  shaft,  should  not  be  less 

than  —  .     Substituting  this  value  of  /  in  equation  (6) 


16 


The  above  would,  therefore,  give  keys  of  breadth  b  =  —  ,  depth 

4 

or  thickness  t  —  j  .6  b  =  .4  d,  and  minimum  length  —  d.     Keys  as 

used  in  practice  conform  closely  to  these  rules  as  far  as  length 
and  breadth  are  concerned;  but,  to  avoid  cutting  away  so  much 
of  the  shaft,  the  thickness  is  usually  much  less  than  that  given 

above.     An  average  value  of  the  thickness  may  be  taken  at  |-  b. 

8 

This  gives  a  key  considerably  thinner  than  it  is  wide  and  makes 
it  weakest  in  crushing.  The  crushing  resistance  can,  however, 
be  increased  by  lengthening  the  key  or  by  using  a  hard  grade  of 
steel. 

Keys  designed  as  above  usually  have  an  excess  of  strength, 
since  the  friction  between  the  shaft  and  the  hub  materially  de- 
creases the  load  actually  brought  upon  the  key.  In  addition,  as 
has  been  pointed  out,  shafts  are  most  usually  designed  for  stiff- 
ness or  angular  distortion,  and  therefore  are  greater  in  diameter 


196 


MACHINE    DESIGN 


than  would  be  required  for  strength  alone.  If  the  key  is  made 
proportional  to  the  shaft  diameter  as  above,  it  must,  therefore, 
have  excess  strength  against  rupture;  and  such  keys  seldom  fail 
unless  subjected  to  severe  shock  or  extraordinary  loads. 

There  are  no  fixed  standards  for  the  dimensions  of  keys, 
various  machine  builders  having  their  own  standards.*  The 
following  table  may  be  taken  as  representing  average  practice 

when  the  length  of  the  key  is  not  less  than  -  d.     If  the  length 

must  be  less  than  this  value,  the  crushing  stress  should  be  com- 
puted, as  it  may  be  necessary  to  use  two  keys. 

TABLE   XI 

DIMENSIONS    OF    FLAT    KEYS    IN   INCHES 


Diam.  of  Shaft  d.  .  .  . 

i 

Ii 

I* 

If 

2 

*i 

3 

3* 

4 

5 

6 

7 

8 

9 

IO 

Breadth  of  Key  6  .... 

1 

5 

T5 

I 

T76 

i 

1 

1 

I 

i 

i* 

if 

i* 

if 

2 

a* 

Thickness  of  Key  t.  . 

A 

T36 

i 

A 

A 

1 

7 

rs 

* 

f 

H 

it 

1 

i 

Ii 

i| 

The  taper  of  sunk  keys  is  usually  about  %"  per  foot  of  length. 

Another  form  of  sunk  key  is  shown  in  Fig.  58.  This  key 
drives  by  compression  or  as  a  strut.  The  keyways  are  more 
difficult  to  cut,  the  keys  more  difficult  to  fit,  and  the  shaft  is  cut 
deeper  than  for  the  common  form.  It  has  been  used  with  great 
success  on  very  heavy  work. 

70.  Feathers  or  Splines.  Sometimes  it  is  desirable  to  have 
the  hub  free  to  slide  axially  along  the  shaft,  but  constrained  to 
rotate  with  it.  In  such  cases  a  feather  or  spline  is  used.  The 
sides  of  the  spline  are  parallel  and  it  may  be  either  fastened 
rigidly  to  the  shaft  or  it  may  move  with  the  hub.  Small  splines 
are  frequently  dovetailed  into  the  shaft  (or  hub),  as  shown  in 
Fig.  59  (a),  while  larger  ones  are  often  held  in  place  by  means 
of  countersunk  screws  (Fig.  59  b),  or  rivets. 

A  common  way  of  securing  the  feather  so  that  it  will  move 


*  See  Kent's  "Mechanical  Engineer's  Handbook,"  page  977. 


KEYS,    COTTERS,    AND    FORCE    FITS 


197 


with  the  hub  is  shown  in  Fig.  60.  Splines  are  subjected  to  a 
shearing  stress  across  the  mid-section  at  the  radius  of  the  shaft, 
and  to  a  crushing  stress  on  the  sides  in  the  same  way  as  sunk  keys. 
Being  fitted  loosely  on  the  top  and  bottom,  they  do  not  produce 
any  friction  between  the  hub  and  the  shaft  and,  therefore,  offer 
much  less  resistance  than  sunk  keys  to  the  rolling  action  im- 
posed upon  them  (see  Art.  69).  This  rolling  action  tends  to 
bring  a  concentrated  crushing  force  at  a  and  b  (Fig.  59  a) ,  if  the 


FIG.  58. 


FIG.  59. 


feather  is  not  rigidly  secured  to  either  the  hub  or  the  shaft.  For 
this  reason,  and  in  order  also  to  provide  ample  wearing  surfaces, 
feathers  are  usually  given  a  greater  radial  depth  than  sunk  keys, 
and  from  their  general  proportions  are  often  distinguished  as 
square  keys.  It  is  evident  that  the  holding  power  of  splines  is 
not  equal  to  that  of  sunk  keys. 

The  following  table  gives  dimensions  of  feathers  which  agree 
with  common  practice : 

TABLE   XII 

DIMENSIONS    OF    FEATHER    KEYS     IN    INCHES 


Diam.  of  Shaft  d  .  .  .  . 

I 

i* 

A 

i* 
1 

if 

2 

2* 

I 

3 
1 

3i 

1 

4 
i 

5 

4 

6 
if 

7 

4 

8 
if 

9 

IO 
2* 

Breadth  of  Feather  b  . 

\ 

T7S 

* 

2 

Thickness  of  Feather  t 

I 

A 

* 

T9* 

i 

I 

I 

i 

l\ 

if 

if 

if 

2 

2* 

2f 

The  length  of  feather  keys  is,  in  general,  greater  than  that  of 
sunk  keys,  for  the  same  size  of  shaft,  in  order  to  reduce  the  bear- 
ing pressure  and  increase  the  wearing  surface  on  the  sides. 


MACHINE    DESIGN 


71.  Cotters.  A  cotter  is  a  form  of  key  used  to  prevent  relative 
sliding  between  two  members.  Fig.  61  shows  a  method  of 
securing  a  piston  rod  to  a  piston  by  means  of  a  cotter.  In  this 
case  the  connection  is  permanent  in  character,  the  cotter  being 
removed  only  when  the  piston  or  piston  rod  is  repaired  or  re- 
newed. In  other  forms  of  cottered  joints  of  this  character  the 
rod  is  not  tapered,  but  is  prevented  from  sliding  into  the  boss  by 
means  of  a  shoulder  or  by  the  cotter  alone.  The  cotter  is  usually 
rectangular  in  section,  but  sometimes  the  edges  are  rounded  so 
as  to  avoid  sharp  corners  in  the  opening  cut  through  the  rod  or 
to  facilitate  machining.  In  light  work  a  taper  pin  of  circular 
section  is  often  used  as  a  cotter.  Fig.  62  shows  an  arrange- 


FIG.  60. 


FIG.  61. 


FIG.  62. 


ment  of  a  gib  and  cotter  (commonly  known  as  a  gib  and  key), 
such  as  is  used  on  the  ends  of  the  connecting-rod  of  steam  engines. 
The  function  of  the  gib  is  to  prevent  spreading  of  the  strap. 
This  arrangement  permits  a  small  amount  of  adjustment  between 
the  strap  and  the  connecting-rod  for  taking  up  wear  on  the  pin 
and  brasses. 

72.  Stresses  in  Cotters.  A  cotter  of  the  form  shown  in  Fig. 
61  is  a  beam  supported  at  the  ends.  The  exact  distribution 
of  the  loading  is  indeterminate,  as  the  bending  of  the  cotter  tends 
to  concentrate  the  load  near  the  points  of  support.  It  is  suffi- 
ciently accurate,  however,  to  consider  the  load  as  uniformly  dis- 
tributed. The  area  of  the  surface  of  the  cotter  where  it  bears 
on  the  rod,  and  also  on  the  hub,  should  be  sufficiently  great  to 
prevent  crushing  of  the  material.  This  indicates  that  the  dianv 


KEYS,    COTTERS,    AND   FORCE   FITS  199 

eter  of  the  hub  should,  for  similar  materials,  be  twice  that  of 
the  rod,  which  is  the  usual  proportion.  The  section  of  the  cotter 
at  the  point  of  support  should  be  great  enough  to  prevent  shearing, 
and  in  many  cases  it  is  sufficient  to  compute  the  section  for  shear 
alone,  neglecting  the  bending  action. 

When  a  cottered  joint  of  this  character  is  made,  the  cotter 
must  be  driven  in  tight  enough  to  prevent  its  backing  out.  This 
is  especially  true  when  the  load  is  a  reversed  one  as  in  the  case 
of  the  steam-engine  piston.  This  induces  an  initial  stress  in  the 
cotter  and  rod,  over  and  above  that  due  to  the  load  P.  The 
conditions,  in  fact,  are  somewhat  similar  to  those  which  exist  in 
screwed  fastenings  (see  Art.  60).  The  initial  stress  due  to  the 
driving  of  the  cotter  cannot  be  accurately  computed,  though  it 
may  be  very  great.  For  this  reason  all  calculations  of  dimensions 
based  on  the  maximum  applied  load  should  be  modified  to  suit 
the  conditions  of  service  and  the  materials  of  which  the  joint  is 
made.  Thus  if  the  rod  be  of  brass  and  the  hub  or  boss  of  steel, 
as  is  common  in  pump  work,  the  proportions  would  be  different 
from  those  employed  if  all  the  materials  were  of  steel  or  of  steel 
and  cast  iron. 

Let  d  =  the  diameter  of  the  rod  where  the  cotter  passes  through. 
1 1     t--=  thickness  of  cotter. 
"    b  =  breadth  of  cotter. 

Then,  in  order  that  the  net  cross-section  of  the  rod  may  be  as 
strong  in  tension  as  the  cotter  and  rod,  where  they  bear  upon 
each  other,  are  in  crushing, 


4 
-  d 


For  a  steel  rod  and  steel  cotter  where  pc  =  .8  pt,  t  =  .44^       .     (2) 

Good  practice  gives  £  =  4  /  =  1.76  d (3) 

The  taper  of  cotters,  as  shown  in  Fig.  61,  should  be  so  small 
that  there  is  no  danger  of  backing  out  and  should  not  exceed  X 
inch  per  foot  of  length.  An  auxiliary  locking  device  is  often 


200  MACHINE    DESIGN 

used  in  arrangements  such  as  shown  in  Fig.  62,  in  which  case 
the  taper  may  be  as  great  as  i  in  8. 

In  the  form  of  cotter  shown  in  Fig.  62,  the  stress  due  to 
driving  the  key  may  be  disregarded,  and  the  design  based  on  the 
maximum  applied  load.  The  student  is  referred  to  treatises  on 
steam-engine  design  for  relative  proportions  of  this  form. 

It  is  often  necessary  to  allow  a  rather  high  bearing  pressure 
on  the  cotter  to  avoid  large  and  clumsy  proportions.  An  ex- 
amination of  successful  practice  shows  an  allowable  pressure  of 
15,000  pounds  per  square  inch  as  computed  from  the  applied 
load. 

FORCE    AND   SHRINKAGE    FITS 

73.  General  Considerations.  Crank  discs,  the  hubs  of  heavy 
fly-wheels,  impulse  water  wheels,  and  work  in  general  which  is 
to  be  subjected  to  shock  or  vibration,  must  be  fastened  to  the 
shaft  more  securely  than  can  be  accomplished  with  a  key,  when 
the  hub  is  a  sliding  fit  on  the  shaft.  In  such  cases  the  bore  of 
the  hub  is  made  slightly  smaller  than  the  diameter  of  the  shaft, 
and  the  shaft  is  forced  cold  into  the  hub;  or  the  hub  is  expanded 
by  heating  till  the  bore  is  slightly  larger  than  the  shaft,  then 
slipped  over  the  shaft  and  allowed  to  cool  in  place.  The  first 
method  is  known  as  a  force  or  pressure  fit,  and  the  second  as  a 
shrinkage  fit.  The  degree  of  tightness  or  "grip"  required  be- 
tween shaft  and  hub  depends  largely  on  the  service.  Thus,  with 
shafts  up  to  three  or  four  inches  in  diameter,  a  difference  between 
the  diameter  of  the  shaft  and  the  bore  such  that  the  parts  may 
be  driven  together  with  a  hand  sledge  is  often  satisfactory.  Such 
a  fit  is  called  a  driving  fit,  and  the  difference  between  the  shaft 
diameter  and  the  bore  is  very  small.  With  such  work  as  arma- 
ture spiders  and  fly-wheel  hubs,  the  allowance  for  the  press  fit 
depends  largely  on  the  facilities  for  erection.  If  the  parts  can  be 
forced  together  in  the  shop,  where  adequate  means,  in  the  form 
of  a  powerful  hydraulic  press  is  to  be  had,  an  allowance  requiring 
a  pressure  of  one  hundred  tons  or  more  may  be  made.  But  if 
the  parts  must  be  erected  in  the  field,  this  allowance  may  have 
to  be  reduced  on  account  of  the  difficulties  of  erection.  It  is 


KEYS,    COTTERS,    AND    FORCE    FITS  2OI 

usually  possible  in  the  case  of  armature  spiders,  fly-wheel  hubs, 
etc.,  to  obtain  a  sufficiently  tight  grip  on  the  shaft  by  means  of 
a  press  fit  without  inducing  undue  stress  in  the  parts.  Depend- 
ence for  preventing  relative  rotation  may  be,  in  a  large  measure, 
placed  upon  the  key  in  all  such  cases. 

In  such  work  as  crank  shafts  when  built  up  from  separate 
parts,  it  is  often  necessary  to  insure  as  strong  a  grip  upon  the 
shaft  as  is  possible  without  inducing  undue  stress.  A  greater 
difference  between  the  shaft  diameter  and  the  bore  of  the  hub  is 
then  allowed  than  in  forced  fits  and  the  parts  are  usually  put 
together  by  shrinking.  In  the  latter  cases  the  stresses  induced 
are  of  importance  and  should  be  carefully  considered. 

74.  Stresses  Due  to  Force  Fits.  If  x  be  the  elongation  or  con- 
traction of  any  radius  r,  then  2  •*  oc  is  the  corresponding  elongation 
or  contraction  of  the  circumference  2  n  r.  The  elongation  or 

2  ~  OC 

contraction  of  the  circumference  per  unit  of  length  is  -     —  .     If 

p  be  the  stress  which  would  induce  this  change  of  length  of 
circumference,  and  E  be  the  coefficient  of  elasticity  of  the  material, 
then 

p  fir 

E  =  -2-,     or    x  =  %-      .....      (i) 


In  Fig.  63,  let  A  represent  a  hollow  shaft  on  which  has  been 
forced  or  shrunk  a  hub  or  boss  B,  the  radius  of  the  contact  surface 
being  r2.  Before  the  operation  of  pressing,  the  outer  radius  of 
the  shaft  was  r2  +  e2,  and  the  inner  radius  of  the  hub  was  r2  —  e'2. 
The  hub  B  is,  therefore,  in  the  condition  of  a  thick  cylinder  sub- 
jected to  an  internal  pressure,  and  the  shaft  A  is  in  the  condition 
of  a  thick  cylinder  subjected  to  an  external  pressure.  The  great- 
est tensile  stress  will  be  found  at  the  inside  surface  of  the  hub, 
and  the  greatest  compressive  stress  at  the  inside  surface  of  the 
shaft.  If,  therefore,  e  be  the  difference  between  the  outer  radius 
of  the  shaft  and  the  inner  radius  of  the  hub,  before  pressing, 
then  e  = 


202  MACHINE    DESIGN 

Let  pt  be  the  unit  tensile  stress  in  the  hub  at  a  radius  r2. 
Let  pc  be  the  unit  compressive  stress  in  the  shaft  at  a  radius  r2. 
Let  w2  be  the  unit  radial  pressure  between  A  and  B. 
Let  r±  be  the  internal  radius  of  the  shaft. 
Let  r3  be  the  external  radius  of  the  hub. 
Then  from  (i) 

'6  =  E*2  +  Er*   °r    A  +  P<  =  ff        "•'      '     (2) 
The  general  equation  for  the  stress  in  thick  cylinders  of  this  kind 

is,— 

4.  T  2   Y  2 

*  v>t  —  2  r22  w2  +  --V-1  (wl  -  w2) 


3         - 

Where  r  j  is  the  inner  radius  of  the  cylinder,  r2  the  outer  radius, 
wl  the  internal  unit  pressure,  w2  the  external  unit  pressure  and 
p  the  tensile  or  compressive  stress  at  any  radius  r.  Applying  this 
equation  to  the  shaft,  wl=o)  r  =  r2,  whence  -the  compressive  stress 
at  the  surface  of  the  shaft  is 

w2  (2  r  22  +  4  r,2) 


In  a  similar  way  substituting  in  the  general  equation  ^  for  r2, 
r2  for  r3,  w1  for  7£/2  and  ze^2  for  w3,  the  unit  tensile  stress  on  the 
inner  surface  of  the  hub  is 

w*  (2  r22  +  3  r2) 


Dividing  (4)  by  (5) 

PC  a 

From  (2)  and  (6) 

Eep 


gg         ...      ....      .     (8) 

2  (  «  +  p)  ' 


*  The  following  treatment  is  from  Professor  Merriman's  "  Mechanics  of 
Materials,"  1906  edition,  page  396.  The  notation  has  been  changed  to  agree  with 
that  adopted  in  this  work. 


KEYS,    COTTERS,    AND    FORCE   FITS  203 

When  the  shaft  is  solid,  r1  in  the  above  equation  becomes  zero 
and  the  equations  are  much  simplified. 

Example.  A  hollow  steel  shaft  10  inches  outside  diameter 
and  2  inches  inside  diameter  is  to  have  a  steel  crank  shrunk  upon 
its  end.  The  hub  of  the  crank  is  18  inches  in  diameter.  What 
must  be  the  difference  between  the  diameter  of  the  shaft  and  the 
bore  of  the  crank  so  that  the  tensile  stress  at  the  inner  surface 
of  the  hub  shall  not  exceed  20,000  Ibs.  per  square  inch?  What 
will  be  the  corresponding  compressive  stresses  at  the  outer  and 
inner  surfaces  of  the  shaft?  Take  £  =  30,000,000. 

Here^  =  i,      r2  =  5,      r3  =  9      and      pt  =  20,000 

Whence  .  -  *-£±*£  -  (°  X        +  «  *  ^  -  * 

4 


V 

2r22 

+  4  rf 

(2  X  S2)  4 

-(4X92) 

2-23 

3W 

'-r,') 

3(92- 

-52) 

and  .?  = 
Then  from  (7) 

Ar,   (a  +  fl        20,000X5(^+2.23) 

£  =  -     '—= =  -  —  =  .0044 

Ep  30,000,000  X  2.23 

From  (8) 

Ee  «  30,000,000  X  .0044  X  .75 

~  r2(  «    +  p)  ~  5  X  2.98 

From  (4) 

PC 


a  3 

4 

and  substituting  this  value  in  (3) ,  making  r  =  r1  and  wl  =  o,  it  is 
found  that  the  compressive  stress  at  the  inner  surface  of  the  shaft 
is  18,500  Ibs.  per  square  inch. 

It  is  evident  that  if  e  be  assumed,  which  is  usually  the  case, 
the  resulting  pressure  and  stresses  can  be  computed.  It  should 
be  noted  that  pt  must  be  well  within  the  elastic  limit  to  prevent 
the  hub  yielding  and  relieving  the  pressure.  It  appears,  as 
pointed  out  by  Professor  Merriman,  that  the  allowances  made  in 
practice  for  force  fits,  induce  stresses  which  should  be  considered 


2O4 


MACHINE    DESIGN 


if  other  stresses  are  to  act  on  the  members.  Thus,  in  the  example 
given,  the  total  allowance  or  difference  between  the  diameter  of 
the  shaft  and  the  bore  of  the  hub  would  be  2  X. 0044  =  .0088; 


and  the  allowance  per  inch  of  diameter  would  be 


.0088 


10 


=  .00088", 


which  is  close  to  average  practice  for  force  fits,  where  .001"  per 
inch  of  diameter  is  often  allowed.     A  somewhat  greater  allowance 


FIG.  64. 


FIG.  65. 


is  generally  made  for  shrinkage  fits,  as  here  the  difficulty  of  forcing 
on  the  hub  does  not  occur. 

75.  Practical  Considerations  in  Force  and  Shrink  Fits.  The 
foregoing  equations,  while  giving  the  probable  stresses  and  radial 
pressure  resulting  from  a  force  or  shrink  fit  made  with  an  allow- 
ance e,  are  limited  in  their  application  to  the  practical  making  of 
force  fits.  There  is,  generally  speaking,  no  difficulty  in  making 
shrink  fits,  with  any  practical  allowance,  as  far  as  getting  the 
parts  together  is  concerned;  although  greater  skill  is  required  in 
handling  shrink  fits  than  force  fits.  In  making  force  fits,  how- 
ever, the  amount  of  pressure  that  can  be  applied  to  the  parts 
is  often  a  controlling  factor.  The  probable  radial  pressure  be- 
tween the  shaft  and  hub  (w2)  may  be  found  as  above,  but  little  is 
known  of  the  coefficient  of  friction  in  such  work,  and  it  is  evident 
that  this  quantity  will  vary  greatly  with  the  character  of  the 
material,  the  finish  of  the  surface,  and  the  lubricant  applied. 
Experimental  data  are  lacking  on  this  point,  hence  it  is  almost  im- 


20  5 

possible  to  estimate  the  resistance  to  slipping  offered  by  force  or 
shrink  fits.  In  general,  shrink  fits  are  superior  to  force  fits 
since  their  surfaces  are  very  dry  and  unlubricated,  while  those 
of  a  force  fit  are  lubricated.  Total  dependence  is,  therefore, 
seldom  placed  on  the  fit  itself,  but  a  key  is  also  used  for  safety. 
Experience  shows  that  the  pressure  required  to  make  a  force 
fit  will  vary  for  any  given  diameter. 

(a)  Directly  as  the  length  of  the  hub 

(b)  Directly  as  the  allowance  e 

(c)  As  some  function  of  the  radial  thickness  of  hub 

(d)  With  the  character  of  the  materials  and  the  finish  of 

the  surfaces. 

It  is  evident  that  a  mathematical  expression  accurately  expressing 
these  relations  would  not  be  practicable,  and  recourse  must  be 
had  to  successful  practice. 

An  allowance  of  .001"  per  inch  of  diameter  will  represent 
average  practice  in  this  country  for  such  work  as  crank  shafts, 
crank  pins,  and  in  general  where  a  tight  fit  is  required.  For 
armature  spiders,  or  fly-wheels,  one-half  this  allowance  is  often 
sufficient.  For  shrink  fits  a  greater  allowance  is  often  made, 
although  the  foregoing  discussion  indicates  that  this  should  not 
be  much  exceeded  considering  the  stresses  induced. 

For  further  information  and  practical  data  the  student  is 
referred  to  the  following : 

Transactions  of  the  American  Society  of  Mechanical  En- 
gineers, Vol.  XXIV. 

"Machine  Design"  by  Forrest  R.  Jones. 

"Machine  Design"  by  W.  L.  Cathcart. 

Machinery,  Vol.  Ill,  No.  9,  May,  1897. 

76.  Thin  Bands  or  Hoops.  If  the  ring  or  band  which  is  forced 
or  shrunk  on  to  a  member  be  thin,  radially,  compared  to  its 
diameter,  the  assumption  can  be  made,  without  appreciable  error, 
that  the  stress  is  uniform  throughout  the  cross-section  of  the  ring. 
'The  change  of  form  in  the  member  on  which  the  band  is  placed 
due  to  compression  is  so  small  in  such  cases  that  it  may  be 
neglected,  and  the  stress  in  the  band  may  be  taken  as  that  due  to 
stretching  it  over  an  incompressible  body.  This  is  practically 


206  MACHINE    DESIGN 

applicable  to  any  ordinary  shape  of  band,  but  rigidly  true  for 
circular  shapes  only.  Thin  bands  of  this  character  are  usually 
shrunk  into  position. 

Example.  A  thin  steel  band  is  to  be  shrunk  on  to  a  casting 
whose  external  linear  dimension  where  the  band  is  to  be  placed  is 
48  inches.  What  must  be  the  length  of  the  inside  face  of  the  band 
so  that  the  stress  per  unit  area  due  to  shrinking  will  be  30,000  Ibs.  ? 
What  will  be  the  area  of  the  cross-section  of  the  band  in  order 
that  the  total  stress  in  the  band  may  be  60,000  Ibs.  ? 

Let  /  =  the  length  of  band  before  shrinking. 

Then  48— /  =  total  amount  of  elongation  of  band. 

,48  —  / 
and  — - —  =  unit  elongation  of  band. 

Whence,  if  E,  the  coefficient  of  elasticity,  be  taken  as  30,000,000, 

unit  stress       30,000 

then,  E  =  30,000,000  =  — r- r-  =     0       . ,  or  /  =  47.95  ins. 

unit  strain       48 — / 

~~T 

The  total  area  of  the  cross-section  of  the  band  will  be  A  =  — — 

30,000 

=  2  square  inches  which  may  be  distributed  in  any  convenient 
proportions. 

If  the  part  on  which  the  band  is  to  be  shrunk  is  circular  in 
form,  the  band  is  in  the  condition  of  a  thin  cylinder  subjected  to 
an  internal  pressure  w  per  unit  area,  where  w  is  the  radial  pressure 
between  the  band  and  the  part  on  which  it  is  shrunk. 

Therefore  by  Art.  78,  wd  =  2  P,  where  P  is  the  total  stress 

2  P 

per  unit  width  of  the  band  or  w  =  —7-.  Thus  in  the  above  prob- 
lem let  the  band  be  shrunk  upon  a  circular  hub  of  diameter  — , 

and  let  the  cross-section  of  the  band  be  y2"  x  4".     Then  P  = 

60,000  2  P       2  X  15,000 

-  =  15,000,  and  w  =  — j-  =  -     — Q —  -  =  1,962    Ibs.    per 
4  d  4° 

square  inch.  * 

The  steel  tires  of  locomotive  driving  wheels  are  usually  shrunk 
on  with  an  allowance  for  shrinkage  of  .001 "  per  inch  of  diameter 
which  gives  .001  inches  elongation  (  )  per  inch  of  circumference. 


KEYS,    COTTERS,    AND    FORCE    FITS  207 

Taking  £  =  30,000,000,  and  considering  the  tire  a  thin  band,  the 
unit  stress  in  the  tire  is 

p  ^  E  A  =  30,000,000  X  .001  =  30,000  Ibs. 

77.  Other  Forms  of  Shrink  Fits.  Many  machine  parts  such 
as  fly-wheel  rims  are  held  together  by  steel  links  or  bands  shrunk 
into  place.  The  theory  outlined  in  the  preceding  article  is  clearly 
applicable  to  these  members,  and  their  dimensions  should  be 
carefully  calculated  so  that  they  will  not  be  overstrained  by  the 
shrinking  alone.  If  such  members  are  so  designed  that  they  will 
be  stressed  up  to  the  elastic  limit  from  shrinkage  alone,  they  are 
liable  to  be  strained  beyond  the  elastic  limit,  when  an  external 
load  greater  than  the  total  shrinkage  stress  is  applied  to  the  parts 
which  they  hold  together,  and  the  link,  taking  a  permanent  set, 
becomes  ineffective.  In  computing  the  dimensions  of  such  links 
allowance  must  sometimes  be  made  for  the  compression  of  the 
parts  held  together,  but  ordinarily  this  is  small  and  may  be 
neglected. 

Occasionally  a  bolt  or  link  is  used  to  reinforce  a  cast-iron 
member  against  tensile  stress.  Thus  in  open  frames,  Fig.  65,  a 
large  bolt  is  sometimes  placed  on  each  side  of  the  throat  as 
shown.  These  bolts  are  usually  put  in  hot  and  allowed  to  cool  in 
place.  As  ordinarily  applied,  the  benefit  derived  from  them  is 
questionable.  If  they  are  designed  and  fitted  so  as  to  put  the 
frame  in  compression  at  A,  an  amount  equal  to  the  tension  induced 
by  the  working  load  P  at  this  same  point,  without  being  themselves 
strained  beyond  the  elastic  limit  when  the  load  is  applied,  then 
no  stress  can  come  upon  the  frame  itself  from  the  force  P.  If, 
however,  the  bolts  and  frame  are  each  to  carry  part  of  the  load, 
care  should  be  exercised  that  the  stress  induced  in  the  bolts  by 
the  initial  load  due  to  shrinking  is  so  low  that  the  additional 
stress  due  to  the  external  load  does  not  raise  this  initial  stress 
beyond  the  elastic  limit,  thus  giving  the  bolts  a  permanent  set 
and  destroying  their  usefulness. 

Let  ^4,  Fig.  64,  represent  a  cast-iron  member  of  uniform 
cross-section  which  is  to  be  reinforced  against  tensile  stress  by  the 
bolt  B.  Suppose,  first,  that  the  nut  is  screwed  up  till  it  just 


2C>8  MACHINE    DESIGN 

bears  firmly  on  the  casting.  If  now  an  external  tensile  load  is 
applied  to  the  casting,  the  bolt  and  casting  will  be  elongated 
the  same  amount  A.  But  the  coefficient  of  elasticity  of  cast 
iron  is  only  about  one-half  that  of  steel.  Hence,  since  p  =  EA, 
the  stress  per  unit  area  in  the  casting  will  only  be  one- half  that 
in  the  steel.  If  2,000  Ibs.  is  the  allowable  unit  stress  in  the 
casting,  4,000  Ibs.  per  unit  area  is  all  that  can  be  thus  obtained 
in  the  bolt.  This  would  lead  to  unnecessarily  large  bolts. 

Suppose,  however,  that  the  nut  is  set  up  till  a  total  compres- 
sive  load  W  is  applied  to  the  cast  iron.  The  bolt  will  be  elongat- 
ed* and  the  casting  compressed,  the  amount  of  elongation  or 
compression  depending  on  the  cross-section  of  the  respective 
members.  The  unit  stress  induced  in  the  bolt  and  casting  will 
also  be  proportional  to  the  area  of  their  respective  cross-sections. 
If  now  an  external  tensile  load  W  is  applied  to  the  bolt,  the 
tendency  is  to  relieve  the  compressive  stress  in  the  casting  and  to 
increase  the  tensile  stress  in  the  bolt.  When  the  load  applied  is 
sufficient  to  elongate  the  bolt  as  much  as  the  casting  was  origin- 
ally compressed,  the  casting  will  be  relieved  of  all  stress.  If  the 
external  load  W  is  applied  to  the  bolt  through  the  casting  itself, 
it  is  evident  that  practically  the  same  result  is  obtained;  and  after 
the  compressive  stress  in  the  casting  is  fully  relieved  any  further 
addition  to  W  induces  a  tensile  stress  in  the  casting  and  still  fur- 
ther increases  the  tension  in  the  bolt.  Usually  the  cross-sectional 
area  of  the  casting  is  very  much  greater  than  that  of  the  bolt. 
Furthermore  the  compressive  stress  induced  in  the  casting  by  the 
initial  load  on  the  bolt  is  usually  very  small  compared  to  the 
tensile  stress  induced  by  the  working  load.  For  these  reasons 
the  compressive  deformation  in  the  casting  can  usually  be 
neglected  without  appreciable  error;  and  the  bolt  may  be 
designed  on  the  basis  of  the  external  load  alone.  (See  Art. 
60,  Case  a.) 

Example.  In  Fig.  65  let  the  section  AB  be  stressed  by  the 
load  P  whose  arm  is  /.  Let  O  be  the  location  of  the  gravity 
axis  of  the  section  AB.  It  is  desired  to  keep  the  stress  at  A  not 

*  See  Art.  59. 


KEYS,    COTTERS,    AND    FORCE    FLTS  209 

greater  than  3,000  Ibs.  per  square  inch.  The  material  is  to  be 
cast  iron.  Let  P  =  60,000. 

"     /  =  moment  of  inertia  of  section  =  4,500. 

"      e  =  io  inches. 
Also  let  the  area  of  the  section  be  200  square  inches.     Then  from 

(P       P  le\ 
—  +  —  —  J 

/  60,000       60,000  X  30  X  io\ 

=   1-      —  +  -  --  )  =  4,300  Ibs.,  and  it  is  desired 

V    200  4,500 

to  reduce  this  to  3,000  Ibs.  by  reinforcing  bolts.  These  rein- 
forcing bolts  serve  the  double  purpose  of  increasing  the  factor 
of  safety  by  reducing  the  fibre  stress,  and  also  of  decreasing  the 
deflection  of  the  frame  at  the  point  where  the  work  is  done.  Let 
these  bolts  be  located  8"  from  O.  Then  the  compressive  stress 
induced  at  A  by  P'  is 

P'        P'X  8  X  io 


4,500 

But  p  —  p'  must  equal  3,000;  therefore 

P'        P'  X  8 


Whence  P'  =  57,000.  This  is  the  total  tensile  load  on  both  bolts, 
when  the  full  working  load  P  is  applied.  If  the  maximum  stress 
at  the  root  of  the  thread  be  taken  at  15,000  Ibs.,  then  the  area  of 

each  bolt  at  the  root  of  the  thread  is  —  —  -  =  i.o  sq.  in., 

2    X    15,000 

which  corresponds  closely  to  a  i^"  bolt.  The  area  of  the 
body  of  a  i^"  bolt,  where  most  of  the  stretching  takes  place, 
is  2.4  square  inches.  Hence  the  working  stress  in  the  body  of  the 

bolt  is    57'°°0    =  1  1,  880  Ibs.  per  sq.  in.     That  portion  of  the 

2   X   2.4 

boss  which  immediately  adjoins  the  throat  is  subjected  to  an  aver- 
age tensile  stress  nearly  equal  to  the  fibre  stress  at  the  surface 
of  the  throat  or  3,000  Ibs.  per  square  inch.  The  upper  and  lower 
portions  of  the  boss  have  little  or  no  tensile  stress  induced  in 
them,  as  a  consideration  of  a  section  such  as  XX  whose  gravity 
axis  is  at  O',  will  show.  It  will  be  reasonable  to  estimate  that 


2IO  MACHINE    DESIGN 

the  stress  in  the  boss  is  equivalent  to  the  full  stress  of  3,000  Ibs. 
per  square  inch  through  14  inches  of  its  length,  the  total  length 
being  19*.  Neglecting  the  compressive  deformation  of  the  boss 
due  to  the  initial  load  from  the  bolts,  the  stress  induced  in  the 
bolt  when  the  stress  in  the  boss  is  3,000  Ibs.,  will  be  3,000 
X2X  14/19  =  4,500  Ibs.  (remembering  that  the  coefficient  of 
elasticity  of  steel  is  twice  that  of  cast  iron).  Whence  the 
initial  stress  in  the  bolt  will  be  11,880  —  4,500  =  7,380.  The 
allowance  for  shrinkage  necessary  to  give  this  initial  stress  will  be 


A=       =          _  = 

E        30,000,000 

The  number  of  threads  per  inch  on  aiK*  bolt  is  5.  Hence  after 

.0046 
the  nut  has  been  set  up  snugly  it  should  be  given  -      -  =  .023 

5 

of  a  turn,  or  should  be  turned  through  360  X  .023=8.2  degrees. 
This  is  most  easily  done  in  the  case  of  large  bolts  by  first  marking 
the  nut  with  reference  to  the  bolt  when  set  up  snug  in  a  cold  state, 
and  then  heating  the  body  of  the  bolt,  if  necessary,  and  rotating 
the  nut  the  desired  amount,  allowing  it  to  cool  in  position. 

It  is  to  be  especially  noted  that  a  very  small  shrinkage  allow- 
ance is  needed  to  induce  a  great  stress  in  the  bolt.  If  too  great 
an  allowance  is  made,  the  bolts  may  be  stressed  beyond  the  elastic 
limit,  and  take  permanent  set  the  first  time  the  external  load 
is  applied.  When  the  external  load  is  again  applied,  a  force  much 
smaller  than  the  total  load  P  will  strain  the  casting  to  the 
point  where  the  bolt  becomes  effective.  The  total  load  P  will 
strain  the  casting  further  than  it  did  originally,  and  even  if 
the  stresses  induced  are  not  sufficient  to  rupture  the  casting,  the 
stiffness  of  the  frame  is  materiallv  decreased. 


CHAPTER  IX 
TUBES,  PIPES,  CYLINDERS,  FLtJES,  AND  THIN  PLATES 

78.  Resistance  of  Thin  Cylinders  to  Internal  Pressure.  If  a 
hollow  circular  cylinder,  whose  walls  are  very  thin  compared  to 
its  diameter,  is  subjected  to  an  internal  bursting  pressure,  a 
tensile  stress  is  induced  in  the  walls.  This  tensile  stress  is  reduced 
near  the  ends  by  the  action  of  the  ends  themselves  which  tend  to 
hold  the  walls  together.  Let  Fig.  66  represent  one-half  of  a  por- 
tion of  a  thin  cylinder  so  far  removed  from  the  ends  that  their 
effect  may  be  neglected. 

Let  w  =  the  unit  internal  pressure 
"    d  =  the  diameter  of  the  cylinder 
"    r  =  the  radius  of  the  cylinder 
"     t  =  the  thickness  of  the  cylinder  walls 
"    />  =  the  unit  tensile  stress  in  the  longitudinal  section 
"   />t  =  the  unit  tensile  stress  in  the  transverse  section 
"    I  =the  length  of  the  part  considered 

Consider  the  half  of  the  cylinder  as  a  free  body,  and  resolve  all 
forces  perpendicular  to  the  cutting  plane.  The  normal  pressure 
on  a  longitudinal  strip  of  length  /  and  width  rdo  iswlr  do.  The 
component  of  this  force  perpendicular  to  the  cutting  plane  is 
w  I  r  do  sin  0.  The  total  pressure  normal  to  this  plane  is 

/7C  f-K 

iv  I  r  do  sin  o  =  w  I  r  I      sin  o  do  =  2wlr  =  wld. 
J    o 

For  equilibrium  this  normal  force  must  equal  the  resisting  stress 
in  the  two  sides  of  the  cylinder.  Hence 

2  pt  I  =  w  I  d 

w  d  2  pt  wd 

or  p  =  —    (i) ;     w  =  —    (2) ;     or   /  =  —    (3) 

In  other  words,  the  unit  longitudinal  stress  in  the  walls  of  a  thin 

211 


212 


MACHINE    DESIGN 


cylinder  is  equal  to  the  product  of  the  diameter  into  the  unit  inter- 
nal pressure,  divided  by  twice  the  thickness  of  the  cylinder  walls, 
and  is  independent  of  the  length  of  the  cylinder. 


FIG.  66. 


FIG.  67. 


If  a  transverse  section  of  the  cylinder  (Fig.  67)  be  considered, 
it  will  be  seen  that  the  total  pressure  on  the  head,  which  tends 


to    cause   rupture   along    a    transverse   section,    is 


,  and 


this  must  be  equal  to  the  intensity  of  the  transverse  stress  pro- 
duced multiplied  by  the  area  of  the  metal  in  such  a  section,  or, 


wd 
*  <h  =  — 

r  t  .  f 

41 


(4) 


(5) 


or*  = 


wd 
4A 


(6) 


A  comparison  of  (i)  and  (4)  shows  the  stress  in  transverse  sections 
to  be  only  one-half  of  that  in  longitudinal  sections.  For  this 
reason  it  is  very  common  practice  to  make  the  circumferential 
seams  of  a  boiler  shell  with  a  single  riveted  joint,  when  the 
longitudinal  seams  are  double  or  triple  riveted. 


*  In  this  discussion  the  mutual  interaction  of  the  longitudinal  and  transverse 
stresses  is  neglected.  If  a  tensile  stress  pi  is  induced  in  a  body,  the  body  contracts 
laterally  as  if  acted  upon  by  a  stress  %pt  acting  at  right  angles  to  the  line  of  action 
of  pi  where  A  is  Poisson's  ratio.  (See  Merriman's  "  Mechanics  of  Materials,"  1906 
edition,  page  359.)  Therefore  the  true  longitudinal  stress  p\  in  the  above  case 
(since  A  equals  £  for  steel)  is 

i  w  d       i  w  d         ou'^ 

*-#—*  -77—  TT  =  '85  17' 

This  gives  a  lower  value  than  equation  (i)  and  hence  the  latter  is  on  the  side  of 
safetv. 


TUBES,  PIPES,  CYLINDERS,  FLUES,  AND  THIN  PLATES     213 

79.  Thin  Spheres.     Since  all  the  meridian  sections  of  a  sphere 
are  the  same  as  the  transverse  section  of  a  cylinder  of  equal 
diameter,  it  is  evident  that  the  stress  in  the  walls  of  a  sphere  is 
given  by  (4).     If  spherical  heads,  of  the  same  thickness  as  the 
shell,  are  placed  on  a  cylinder  which  is  to  withstand  internal 
pressure,  they  will  be  subjected  to  a  maximum  stress  equal  to  the 
transverse  stress  in  the  shell. 

80.  Resistance  of  Non-Circular  Thin  Cylinders  to  Internal 
Pressure.     Suppose  a  cylinder  to  have  a  cross-section  made  up  of 
circular  arcs  as  in  Fig.  68.     Take  the  upper  half  as  a  free  body 
(section  along  the  major  axis).     Let  the  resultants  of  the  com- 
ponents of  pressure  which  are  normal  to  the  plane  of  the  section 
be  WD  W2,  Ws,  for  the  portion  marked  I,  II,  III,  respectively. 
Then  these  resultant  forces  per  unit  of  length  of  the  cylinder  are 
as  follows: — 

/y 

W1  =  w  r  J      sin  <p  d  <?  =?  w  r  (  —  cos  o  +  cos  <?')  =  w  ml 

f* 

W2  =wR  J      sin  0  d  0  =  w  R  (—  cos  0'  +  cos  0"}  =  w  m2 

/- 
tt  sin  (f>  d  <f>  =  w  r  (  —  cos  r.  +  cos  <f>")  =  wm3 

Therefore  W1  +  W2  +  W3  =  w  (m^  +  m2  +  ms)  =  w  A 

In  a  similar  way,  if  the  section  is  taken  along  the  minor  axis, 
the  resultant  force  normal  to  this  axis  is  found  to  be  wB.  In  like 
manner  the  resultant  force  normal  to  any  section  is  (per  unit  of 
length  of  cylinder)  equal  to  the  intensity  of  pressure  multiplied 
by  the  axis  of  that  section.  As  B  is  less  than  A,  the  resultant 
force  wB  is  less  than  wA;  or  the  force  tending  to  elongate  the 
minor  axis  is  greater  than  the  force  tending  to  elongate  the  major 
axis.  If  the  tube  were  perfectly  flexible,  its  form  of  cross-section 
would  become,  under  pressure,  one  in  which  all  axes  are  equal, 
or  circular.  A  rigid  material  offers  resistance  to  such  change  of 
form,  and  a  flexural  stress  is  produced  in  addition  to  the  direct 
tension,  but  it  approaches  nearer  to  the  circular  form  as  the  pres- 
sure increases.  The  existence  of  this  flexural  stress  in  a  non- 
circular  cylinder  becomes  apparent  from  a  comparison  of  Figs.  69 


214 


MACHINE    DESIGN 


and  70.  In  Fig.  69  (circular  section)  the  lines  of  normal  pressure 
all  pass  through  a  single  point  (the  centre  of  the  circle) ;  the  re- 
sultant (Pr)  of  the  tensions  (Pt  and  P2)  also  passes  through  this 
same  point,  hence  these  forces  form  a  concurrent  system,  and 
they  are  in  equilibrium.  In  Fig.  70,  however,  the  pressures  do 
not  in  themselves  form  a  concurrent,  nor  parallel,  system  of  forces, 
hence  they  cannot  be  balanced  by  a  single  force  (as  the  resultant 
Pr),  but  there  must  be  a  moment,  or  moments,  of  stress  for  equi- 
librium. A  similar  course  of  reasoning  could  be  applied  to  a 
cylinder  of  any  non-circular  cross-section;  for  such  a  section 
(Fig.  71)  could  be  considered  as  made  up  of  circular  arcs,  each  of 
which  could  be  treated  (like  the  special  case  of  Fig.  68)  by  inte- 


FIG.  68. 


FIG.  71. 


grating  between  proper  limits.  A  direct  inspection  will  also  show 
that  in  any  non-circular  section  cylinder,  subjected  to  internal 
pressure,  the  pressure  tends  to  reduce  the  cylinder  to  a  circular 
cross-section.  Suppose  the  original  cylinder  (Fig.  71)  to  be  cut 
along  the  greatest  axis  of  its  cross-section,  and  that  a  flat  bottom 
coinciding  with  this  section-plane  be  secured  to  it,  the  lower 
portion  of  the  cylinder  being  entirely  removed.  The  total 
pressure  on  this  bottom  evidently  balances  the  components  of 
the  pressure  on  the  curved  surface  which  lie  normally  to  this  flat 
bottom;  hence,  the  resultant  of  these  normal  components  of  pres- 
sure equals  w  (a  .  .  a)  =w  A,  per  unit  of  length  of  cylinder. 
In  a  similar  way,  the  resultant  of  components  of  pressure  acting 
normally  to  any  other  section  (as  b  .  .  b,  Fig.  71)  equals 


TUBES,  PIPES,  CYLINDERS,  FLUES,  AND  THIN  PLATES       215 

w  (b  .  .  b)=wB<wA.  This  direct  method  might  have 
been  used  in  the  preceding  cases  (Figs.  66  and  68)  without  re- 
course to  the  calculus. 

It  is  apparent,  then,  that  any  cylinder  under  internal  pressure 
tends  to  assume  a  circular  cross-section.  A  cylinder  of  nominal 
circular  section,  but  departing  from  the  true  form  to  some  extent, 
tends  to  correct  this  departure  under  internal  pressure;  or  if  a 
circular  cylinder  under  internal  pressure  is  deformed  by  any  ex- 
ternal force,  it  tends  to  resume  its  circular  shape.  Thus  a  circular 
cylinder  under  internal  pressure  is  in  "stable  equilibrium."  If 
the  section  is  other  than  a  true  circle  there  is  a  flexural  stress,  as 
well  as  tension,  when  under  pressure. 

RESISTANCE  OF  THIN  CYLINDERS  TO  EXTERNAL 
PRESSURE 

81.  Theoretical  Considerations.  If  a  thin  hollow  cylinder  of 
circular  section  is  subjected  to  an  external  pressure,  it  is  obvious 
that  a  course  of  reasoning  similar  to  that  in  Art.  78  will  show 
that  a  compressive  stress  is  induced  in  the  walls  of  the  cylinder, 
the  value  of  which  will  be  given  by  formula  (i)  Art.  78  or 

w  d 
p  =  —  where  p  is  a  compressive  stress. 

If  the  cylinder  were  perfectly  cylindrical,  of  uniform  thickness, 
and  of  homogeneous  material,  there  seems  to  be  no  reason  why 
failure  should  occur  until  the  compressive  stress  reaches  the  yield 
point  of  the  material.  But  tubes  are  never  absolutely  circular  in 
form,  uniform  in  thickness,  or  homogeneous  in  character;  and 
hence  failure  occurs  long  before  the  compressive  yield  point  is 
reached.  A  tube  which  fails  under  external  pressure  is  said  to 
collapse,  and  the  forms  of  collapsed  tubes  are  very  characteristic. 
Fig.  72  shows  the  form  of  cross-section  of  collapsed  tubes,  and 
Unwin*  has  shown  that  the  number  of  lobes  depends  on  the 
ratio  of  length  to  diameter,  the  smaller  this  ratio  the  greater  being 
the  number  of  lobes.  This  peculiarity  is  undoubtedly  due  to  the 

influence  of  the  heads  placed  in  the  ends.     For  values  of  -j 

*See  "  Elements  of  Machine  Design,"  page  101,  1901  edition. 


2l6 


MACHINE    DESIGN 


greater  than  about  4  to  6,  only  the  forms  of  collapse  shown  at 
c  and  d,  Fig.  72,  appear. 

If  the  non-circular  cylinders  of  either  Fig.  68  or  71  be  con- 
sidered as  subjected  to  external  pressure,  the  force  tending  to 
increase  the  major  axis  will  be  seen  to  be  greater  than  that  tend- 
ing to  increase  the  minor  axis;  hence  the  external  pressure  will 
cause  collapse,  unless  the  flexural  rigidity  of  the  material  is 
sufficient  to  prevent  this  action.  In  a  cylinder  of  nominal  cir- 
cular section  any  departure  from  the  ideal  section  will  be  increased 
by  the  external  pressure.  Or,  if  a  cylinder  of  true  circular 
section  is  deformed  in  any  way  while  under  external  pressure, 
this  pressure  will  tend  still  further  to  increase  the  deformation. 


FIG.  72. 


FIG.  73. 


In  other  words,  a  cylinder  under  external  pressure  is  in  "  unstable 
equilibrium."  As  perfectly  true  circular  sections  and  homogene- 
ous materials  are  not  attainable  in  practice  the  danger  of  collapse 
must  be  taken  into  consideration  in  designing  pipes,  tubes,  or  flues 
to  withstand  external  fluid  pressure. 

Since  the  wall  of  an  ideal  thin  tube  is  subjected  to  a  uniform 
compressive  stress,  it  may  be  considered  as  being  in  the  same 
condition  as  a  long  column;  and  theoretical  equations  expressing 
the  relation  between  the  external  pressure,  the  stress,  and  the 
dimensions  of  the  tube  have  been  developed  on  this  basis.  In 
view  of  the  fact  that  the  theory  of  long  columns  is  itself  most  un- 
satisfactory, it  is  not  surprising  that  such  equations  do  not  accord 
with  actual  results,  and  they  may  be  safely  disregarded,  but  the 
analogy  between  long  compression  members  and  tubes  sub- 
jected to  external  pressure  is  instructive.  Other  deductions 
based  upon  the  theory  of  elasticity,  while  throwing  some  light 


TUBES,  PIPES,  CYLINDERS,  FLUES,  AND  THIN  PLATES     217 

on  the  form  of  rational  equations  expressing  these  relations,  are 
not  as  yet  applicable  to  practical  problems. 

82.  Long  Tubes,  Pipes,  etc.  Until  very  recently  the  only 
experimental  results  on  the  collapse  of  thin  tubes  were  those  due 
to  Sir  William  Fairbairn,  who,  in  1858,  made  a  series  of  careful 
experiments  on  short  tubes  and  deduced  therefrom  the  following 
formula : 

J  2. 19 

w  =  9,675,600  — (i) 

where  w  is  the  unit  external  collapsing  pressure  in  pounds  per 
square  inch  and  /,  /,  and  d  are  the  thickness,  length,  and  outside 
diameter  respectively  in  inches.  Fairbairn  himself  modified  this 
equation,  for  simplicity,  to  the  form 

t2 
w  -=  9,675,600  —......      (2) 

Many  other  equations  have  been  deduced  from  the  experiments  of 
Fairbairn,  usually  of  the  same  form  but  with  different  exponents. 
Thus  Professor  Unwin  gives  the  following  as  the  result  of  a  careful 
resume  of  Fairbairn's  work: 

For  tubes  with  a  longitudinal  lap-joint 

w  =  7,363,000  j^-^     ....  (3) 

For  tubes  with  a  longitudinal  butt-joint 

w  =  9,614,000  ^9^ii6     ....  (4) 

For  tubes  with  longitudinal  and  cross  joints  like  an  ordinary 
boiler  flue 

^2.35 
w  =  15,547,000^-^. (5) 

Other  writers  have  deduced  similar  equations  from  the  same  data. 
Fairbairn's  experiments  were  conducted  with  tubes  whose 
lengths  were  small  compared  to  their  diameters.  In  such  tubes 
the  effect  of  the  supporting  action  of  the  head  is  noticeable;  hence 
his  equations  make  the  allowable  pressure  vary  inversely  as  some 
function  of  the  length.  Now  it  is  reasonable  to  suppose  that  if 


2l8  MACHINE    DESIGN 

the  tube  were  long  enough  the  head  would  have  no  effect,  except 
near  the  ends,  and  the  collapsing  pressure  would  be  independent 
of  the  length.  In  a  similar  way  if  the  tube  were  very  short,  the 
walls  should  theoretically  yield  by  crushing,  and  the  intensity  of 

w  d 
the  compressive  stress  would  be  given  by  formula  (i)  or,  p  = . 

2  t 

In  1906  Professor  A.  P.  Carman  published*  the  results  of  a 
set  of  experiments  made  at  the  Engineering  Experiment  Sta- 
tion of  the  University  of  Illinois,  which  prove  conclusively 
that  Fairbairn's  equations  hold  only  for  tubes  whose  lengths 
are  from  four  to  six  times  their  diameters;  and  that  be- 
yond that  ratio  the  collapsing  pressure  is  independent  of 
the  length.  He  found  that  the  results  of  his  experi- 
ments could  not  be  well  expressed  by  a  single  equation,  but 
devised  two  equations  to  cover  the  range;  these  equations  ex- 
pressing the  relation  which  exists  between  w  and  — .  Thus  for 
values  of  .-\.O25,  and  length  greater  than  4  to  6  times  the 

diameter,  he  gives  w  =  k  -  —  c  where  k  and  c  are  constants  to 

be  determined  experimentally  and  depending  upon  the  material. 
For  brass  tubes 

™  =  93>365  ~d  ~  2>474 (6) 

For  seamless  drawn  cold  steel 

w  =  95,520  -  -  2,090 (7) 

For  lap-welded  steel 

w  =  83,270—  —  1,025 (8) 

d 

Professor  R.  T.  Stewart,f  in  an  elaborate  set  of  experiments 

*  See  Bulletin  of  the  University  of  Illinois  Engineering  Experiment  Station 
Vol.  Ill,  No.  17,  June,  1906. 

t  See  Transactions  of  American  Society  of  Mechanical  Engineers,  Vol.  XXVII, 
1906. 


TUBES,  PIPES,  CYLINDERS,  FLUES,  AND  THIN  PLATES     2IQ 

on  lap-welded  steel  boiler  tubes  made  for  the  National  Tube 

t  v 
Company,  found  that  for  values  of  —  \  .023  the  results  of  his 

work  could  be  expressed  by  the  following: 

w  =  86,670-^  -  1,386    .....      (9) 

which  corresponds  closely  with  (8)  of  Professor  Carman's  work, 
showing  the  accuracy  of  the  experimental  work. 

For  values  of  -  <.025    Professor    Carman    found    that    the 
results  of  his  work  could  be  expressed  by  an  equation  of  the  form 


where  k'  as  before  is  an  experimentally  determined  constant, 
whose  value  for  thin  brass  tubes  is  25,150,000,  and  for  thin  cold- 
drawn  seamless  steel  tubes  50,200,000. 

Professor  Stewart  found  that  for  values  of  —  below  .023,   or 

practically  the  same  limit  as  above,  his  results  were  expressed  by 

w  =  1,000  (i  --  V  i  —  1,600  -^-  j      .      .      .      (n) 

The  value  of  w  i  or  —  =  .023  is  about  600  Ibs.,  which  corresponds 
closely  with  the  upper  limiting  value  of  w  obtained  from  (io). 

For  values  of  -  less  than  .023,  the  corresponding  values  of 
d> 

w,  as  found  by  either  (io)  or  (n),  do  not  differ  materially. 
Furthermore,  tubes  in  which  —  <^.O2  are  not  much  used  in  en- 

gineering work  under  external  pressure,  and  for  convenience 
therefore  equation  (io)  will  be  adopted. 

83.  Summary  of  Equations  for  Long  Tubes.  The  works  of 
Stewart  and  Carman  deal  entirely  with  tubes  which  are  so  long 
that  the  supporting  effect  of  the  heads  is  negligible,  or  in  which  the 
length  is  at  least  four  times  the  diameter.  Their  experiments,  while 


220  MACHINE    DESIGN 

conducted  separately,  supplement  and  corroborate  each  other. 
As  given  above,  the  equations  are  not  in  the  most  convenient 
form  for  use  by  the  designer,  since  usually  /,  d  and  w  are  known 
and  /  is  required.  Transposing  these  equations,  therefore,  they 
may  be  written  as  follows: 

For  values  of  —  ^.023  and  pressures  less  than  600  equation 
10  becomes 

3   I 

/  =     \T      "     "    '     "     '     *    ^ 

where  £  =  25,150,000  for  thin  brass  tubes,  and  50,200,000  for  thin 
cold-drawn  seamless  tubes  or  lap-welded  steel  tubes. 

For  values  of  -^>  .023  and   pressures  greater  than  600  Ibs., 
equation  6  becomes 


where  for  brass  tubes     .      .      .      .     £  =  93,365  and  c  =  2,  474 
"       "  seamless  cold- 

drawn  steel  .  .  .  £  =  95,520  and  c  =  2,090 
"  "  lap-welded  steel  .  ,  £  =  83,270  and  c  =  1,025 
The  following  approximate  formula,  which  covers  practically 

the  whole  range  of  values  of  —  ,  is  suggested  by  Professor  Car- 
man as  useful  in  making  rough  calculations. 


....  (i3a) 

where  £"  =  1,000,000  for  cold-drawn  seamless  tubes  and  1,250,000 
for  lap-welded  steel  tubes.  From  this,  the  following  usually 
more  convenient  formula  can  be  derived: 

V    \.S  t.^  .     .|j|    .     .     (I3b) 

Example.  A  lap-welded  steel  boiler  tube  4  inches  outside 
diameter  and  10  feet  long,  is  subjected  to  an  external  pressure  of 
300  pounds  per  square  inch.  What  must  the  thickness  be  in 
order  to  have  a  factor  of  safety  of  at  least  6  ? 


TUBES,  PIPES,  CYLINDERS,  FLUES,  AND  THIN  PLATES      221 

Here  the  assumed  collapsing  pressure  is 

300  X  6  =  i,  800  Ibs.  per  square  inch. 
Applying  equation  (13) 

d(w  +  c)       4(1,800+1,025) 
t  =  -  -  -  =  -  —  =  .14  inch. 

k  83,270 

/        .14 
Here  the  ratio  -  =  -  =  .035,  and  hence  equation  (13)  applies. 

In  case  this  ratio  should  be  less  than  .023,  which  will  seldom 
occur,  a  second  solution,  using  equation  12,  should  be  made. 
84.  Short  Cylinders,  Flues,  etc.     When  the  cylinder  or  flue  is 


short,  i.e.,  ~j\4  to  6,  the  effect  of  the  heads  should  not  be  neg 

lected  as  in  Carman's  and  Stewart's  work,  and  Fairbairn's  ap- 
proximate equation  is  applicable,  or 


=  9,675,600^-,  ......      (14) 


or  transposing 

" 


If  a  cylinder  under  external  pressure  could  be  depended  upon 
to  fail  only  by   actual  crushing,   instead  of    through    collapse 

(buckling),   then   the   formula  w  =  —  J-  —  would  apply,  as  in 

internal  pressure;  remembering  that  under  external  pressure 
the  stress  p  is  compression.  If  this  equation  gives  a  lower  work- 
ing pressure  than  (14)  the  flue  designed  by  it  will  be  safe  against 
collapse.  The  rules  of  the  Lloyd's  Marine  Register  allow  the 

following   pressure   in    boiler   flues:     w=  —  —  ^—.  --  .     This   is 

/    (i 

Fairbairn's  equation  with  a  factor  of  safety  of  9.  The  British 
Board  of  Trade  rules  allow  a  working  stress  in  furnaces  and  flues 

2  p  t 
of  about  4,000  when  computed  by  the  equation  w  =  ~~r~-    This 


222  MACHINE    DESIGN 

is  a  little  less  than  that  allowed  by  the  U.  S.  Board  of  Supervising 
Inspectors. 

Hence  for  the  same  allowable  pressure  under  these  two  rules 

2pt         8,000/          1,075, 200/2 

w"  ~T     ~T        ~iT 

or/  m  134.4* 

If,  therefore,  I  <  134.4  t  equations  (i),  (2),  and  (3)  may  be  safely 
used. 

It  will  be  observed  that  this  relation  limits  the  use  of  these 
equations  to  comparatively  short  flues.  Thus  a  flue  %"  thick 
could  only"  be  34"  long  to  have  these  equations  applicable.  In 
practice  long  flues  of  large  diameter  are  reinforced  at  short  inter- 
vals by  heavy  rings  of  rolled  or  other  section,  known  as  collapse 
rings,  thus  making  the  flue  consist  virtually  of  a  series  of  short 
flues,  to  which  equations  (i),  (2),  and  (3)  may  be  applied. 

Various  Insurance  and  Government  inspection  departments 
give  rules  for  proportioning  flues  and  furnaces.  These  rules 
change  from  time  to  time,  and  if  the  boiler  is  to  be  insured  in 
any  company  the  specific  rules  prescribed  by  it  should  be  consulted. 

Thus  Lloyd's  Register  for  1906-7  gives 

1,075,200  t2 
w  = ,       . when  /   >  120  t 

I  /\  CL 

50  (  300  /  —  /) 

and  w  =  —  -  when  /  <  120  t 

d 

where  /,  /  and  d  are  all  in  inches. . 

Various  other  authorities  give  similar  equations  with  prac- 
tically the  same  coefficients. 

85.  Corrugated  Furnace  Flues.  Flues  corrugated  as  in  Fig. 
73  are  very  much  stiffer  against  collapse  than  plain  cylindrical 
flues,  and  with  proper  dimensions  of  corrugations  may  be  safely 
made  of  any  desired  length.  Their  peculiar  shape  also  permits 
of  expansion  and  contraction  under  the  influence  of  heat.  When 
the  corrugations  are  not  less  than  i  %  inches  deep,  and  not  more 
than  8  inches  from  centre  to  centre  of  corrugations,  and  plain 


TUBES,  PIPES,  CYLINDERS,  FLUES,  AND  THIN  PLATES      223 

portions  at  the  ends  do  not  exceed  9  inches,  the  U.  S.  B.  S.  I. 
allows  a  working  pressure  of 

14,000  t 


This  is  also  the  formula  of  the  British  Board  of  Trade. 

Lloyd's  Register  for  1907-8  gives  a  number  of  rules  for  design- 
ing various  types  of  flues. 

The  following  references  contain  valuable  practical  informa- 
tion on  this  subject  : 

Lloyd's  Register  of  British  and  Foreign  Shipping. 

"  Steam  Boilers,"  by  Peabody  and  Miller. 

Rules  and  Regulations  of  U.  S.  Board  of  Supervising  In- 
spectors. 

Rules  and  Regulations  of  the  American  Bureau  of  Shipping. 

Rules  of  the  British  Board  of  Trade. 

Rules  of  the  Bureau  Veritas/ 

Seaton  and  Rounthwaite's  Pocket  Book. 

THICK   CYLINDERS 

86.  When  the  wall  of  a  cylinder,  which  is  subjected  to 
internal  or  external  fluid  pressure,  is  thick  relatively  to  the 
internal  diameter,  it  can  no  longer  be  assumed  that  the  stress 
in  the  wall  is  uniformly  distributed  over  the  cross-section,  but 
it  is  greater  at  the  inner  surface  and  decreases  to  a  minimum 
at  the  outer  surface  whether  the  pressure  is  internal  or  external. 
When  the  pressure  is  internal  the  stress  is  tensile,  and  when  the 
pressure  is  external  the  stress  is  compressive. 

Many  formulae  have  been  'deduced  to  express  the  relations 
between  pressure,  stress,  and  cylinder  thickness.  Of  these,  that 
of  Lame,  deduced  in  1833,  is  perhaps  best  known.  Clavarino's* 
modification  of  Lame's  formula,  which  was  published  in  1880,  is 
now  much  used  and  will  be  adopted  in  this  work. 

Ordinarily  the  cylinder  is  subjected  to  either  external  or 
internal  pressure  alone;  but  in  a  gun  tube,  for  example,  which 

*  The  student  is  advised  to  read  the  discussion  of  thick  cylinders  given  in  Merri- 
man's  "Mechanics  of  Materials,"  edition  of  1906. 


224  MACHINE    DESIGN 

has  a  hoop  shrunk  upon  it,  the  more  general  case  occurs  in  which 
the  cylinder  is  subjected  to  both  internal  and  external  pressure. 
Let  wl  =  the  internal  unit  pressure. 

"  w2  =  the  external    "         " 

"  fj  =  the  internal  radius  of  the  cylinder. 

"  r2  =  the  external     " 

"  pl  =  the  unit  stress  at  the  inner  surface. 

"  />2  =  the    "         "         "       outer      " 
Then  by  Clavarino's  equation  the  unit  stress  at  any  radius  r  is 


If  the  external  pressure  w2  be  zero,  which  is  the  most  usual 
case,  the  greatest  tensile  stress  is  at  the  inner  surface,  and  is 


3ft  —  4 

Example.  A  cast-iron  cylinder  20  inches  in  internal  diameter. 
is  to  withstand  an  internal  pressure  of  1,000  Ibs.  per  square  inch. 
How  thick  must  the  wall  be  in  order  that  the  stress  at  the  inner 
surface  may  not  exceed  4,000  Ibs.  per  square  inch  ? 

Here  r  =10,  wl  =  iyooo  and  p1  =  4,000.     Hence  substituting 
in  (18) 
r   =  r   r  3  ft  +  ^i  "I  1  _  IQ  F     3  X  4,000  +  1,000     -.  =  ^  g/, 

1  1-3  ft  -4  ^J  L3  X  4,000  —  4  X  1,000  J 

or  the  cylinder  walls  must  be  2.8"  thick. 

From  (16)  it  is  found  that  p2,  the  stress  in  the  cylinder  walls 
at  the  outer  fibre,  is  2,620  Ibs. 

PRACTICAL  CONSIDERATIONS 

87.  Cast-iron  pipes  are  widely  used  for  underground  water 
pipes  and  to  some  extent  also  for  gas  pipes,  largely  on  account 
of  their  durability  against  corrosion.  For  steam,  or  for 
high  pressures  generally,  cast-iron  pipes  are  now  seldom  used 


TUBES,  PIPES,  CYLINDERS,  FLUES,  AND  THIN  PLATES     225 

because  of  their  unreliability.  For  all  ordinary  purposes 
pipes  made  of  wrought  iron  or  steel  are  most  used,  although 
in  special  cases,  such  as  marine  work,  copper  and  brass  are 
preferred. 

Wrought-iron  or  steel  pipes  may  be  either  lap-welded  or  butt- 
welded,  the  latter  being  commonly  used  for  the  smaller  diameters, 
while  steel  piping  may  be  "drawn"  so  that  there  is  no  seam,  in 
which  case  it  is  known  as  "seamless  drawn  tubing." 

Standard  Piping  is  designated  by  its  nominal  internal  diameter. 
Thus  standard  i-inch  gas  pipe  has  a  nominal  internal  diameter 
of  i  inch,  and  an  external  diameter  of  1.315  inches.  So-called 
standard  wrought-iron  piping  may  be  used  for  pressures  up  to 
100  Ibs.  with  safety.  For  still  higher  pressures,  such  as  are  found 
in  high-class  steam  plants,  thicker  pipes,  known  as  extra  strong, 
are  used.  For  hydraulic  work,  where  pressures  up  to  several 
thousand  Ibs.  per  square  inch  must  be  withstood,  still  thicker 
piping,  known  as  double  extra  strong,  is  used.  These  heavy  pipes 
are  made  by  decreasing  the  internal  diameter  of  the  standard 
pipe,  thus  keeping  the  outside  diameter  and  hence  the  screw 
threads  for  the  flanges  to  one  standard.*  Thus  an  extra  strong 
i-inch  pipe  (nominal  size)  would  have  an  internal  diameter  of 
.95  inches,  and  a  double  extra  strong  of  the  same  nominal  size 
would  have  an  internal  diameter  of  .587  inches,  the*  external 
diameter  remaining  1.315  inches  in  all  cases. 

For  large  cylinders  both  for  steam  and  hydraulic  service, 
cast  iron  is  still  much  used  and  probably  will  be  for  some  time 
yet,  on  account  of  the  ease  with  which  complicated  iron  castings 
can  be  made  and  machined.  In  the  case  of  steam-engine  cylin- 
ders the  thickness  of  the  walls  is  fixed  by  considerations  other 
than  those  of  strength,  such  as  stiffness  and  securing  good  cast- 
ings. The  proportions  of  steam  cylinders  as  fixed  by  practice 
are  the  best  guide.  An  examination  of  current  practice  shows 
the  average  thickness  of  low-speed  engines  to  be  given  by  the 


*  The  student  is  referred  to  Kent's  "  Engineer's  Pocket  Book,"  or  similar  works, 
for  full  tables  of  standard  sizes  of  pipes,  flanges,  etc.     See   also  current  trade 
catalogues. 
15 


226  MACHINE    DESIGN 

following,  /  =  . 05*2 +  .3  inch,*  where  /  =  thickness  and  d  =  diameter 
in  inches,  when  the  steam  pressure  does  not  exceed  125  Ibs.  per 
sq.  inch. 

Kent's  "  Mechanical  Engineer's  Pocket  Book"  gives  the  follow- 
ing as  representing  current  practice,  t  =  .ooo4dp  +  .3,  where  d  = 
diameter  in  inches  and  p  =  pressure  in  pounds  per  square  inch. 
If  p  be  taken  as  125  pounds  this  equation  reduces  to  that  given 
by  Barr< 

Cast  iron  is  also  much  used  for  the  cylinders  of  hydraulic 
machines,  although  steel  castings  are  better  in  general.  In  such 
cases  equations  (16)  to  (18)  developed  above,  in  common  with  all 
equations  based  on  the  theory  of  elasticity,  should  be  used  with 
caution  when  cast  iron  is  selected  for  the  cylinder.  Further- 
more it  must  be  borne  in  mind  that  the  thicker  the  cylinder  walls, 
the  more  liable  are  they  to  be  porous  in  the  interior,  where  made 
of  castings.  It  is  safer,  therefore,  as  a  rule,  to  carry  a  high  work- 
ing stress,  within  safe  limits,  and  insure  sound  castings,  than  to 
design  thick  walls  which  are  open  to  suspicion,  in  order  to  get  a 
theoretically  lower  stress.  A  3 -inch  wall,  for  instance,  with  a 
working  stress  of  5,000  pounds  per  square  inch  is  preferable  to  a 
4-inch  wall  with  a  working  stress  of  3,000  pounds  per  square  inch. 
Care  should  also  be  exercised  in  cylinders  made  of  castings  to 
avoid  excessive  thickness  of  metal  at  any  point,  thus  insuring 
sound  castings.  Thick  castings  of  any  metal  are  very  liable  to 
give  trouble  by  leaking  on  account  of  porosity,  if  subjected  to  high 
pressures,  and  cast-iron  cylinders  are  often  lined  with  brass  or 
bronze  liners  to  obviate  this  difficulty. 

87.1.  Pipe  Couplings,  Flanges,  etc.  Methods  for  securing 
the  ends  of  pipes  together  have  become  of  greater  importance 
as  higher  steam  pressures  have  been  employed.  The  most  usual 
method  for  accomplishing  this  purpose  has  been  to  thread  the 
ends  of  the  pipes  (see  Art.  57)  and  secure  them  together  with 
either  a  cylindrical  pipe  coupling,  a  pipe  union,  or  a  pair  of  pipe 
flanges.  All  of  these  are  in  very  common  use.  For  pressures 
up  to  100  pounds  per  square  inch  and  pipes  not  over  12  inches  in 

*See  "Current  Practice  in  Engine  Proportions,"  by  J.   H.  Barr;   Transac- 
tions A.  S.  M.  E.,  Vol.  XVIII. 


TUBES,  PIPES,  CYLINDERS,  FLUES,  AND  THIN  PLATES       227 

diameter  these  may  be  used  with  success,  but  for  higher  pressures 
and  larger  diameters  they  are  not  so  satisfactory.  The  union 
is  used  on  small  pipes  only. 

In  the  ordinary  screwed  fitting  of  large  size  it  is  difficult  to 
cut  the  thread  accurately,  and  to  screw  the  fitting  on  tight 
enough  to  prevent  leakage  at  A,  Fig.  73  (a).  This  can  be  rem- 
edied to  some  extent  by  making  the  threaded  portion  of  the  pipe 
long  enough  to  project  through  the  flange  slightly,  and  then 
facing  off  pipe  and  flange  so  as  to  make  a  smooth  surface,  and 
permitting  the  packing  or  gasket  (P)  to  cover  up  the  screwed 
joint,  as  shown  at  B,  Fig.  73  (a).  Even  this  joint,  however,  is 
liable  to  leak  if  the  workmanship  is  poor  or  if  the  flanges  do  not 
align  properly. 

To  obviate  the  difficulties  of  the  screwed  joint  on  pipe  of 


FIG.  73  (a). 


FIG.  73  (b). 


FIG.  73  (c). 


larger  diameter,  the  flanges  are  sometimes  shrunk  on  as  shown 
in  Fig.  73  (b)  (see  also  Art.  73).  In  order  to  insure  tightness, 
and  secure  a  firmer  grip  on  the  flange,  the  end  of  the  pipe  is 
usually  expanded  into  the  flange,  as  shown  in  Fig.  73  (b).  The 
gasket  usually  covers  up  the  joint  between  the  pipe  and  the 
flange.  This  form  of  coupling  is  not  well  suited  for  high-pressure 
work,  however,  especially  if  the  pipe  is  not  machined  on  the  out- 
side before  the  flange  is  shrunk  on.  When  subjected  to  the 
heavy  straining  action  incident  to  expansion  and  contraction, 
as  in  heavy  steam  mains,  the  pipe  is  sure  to  work  in  the  flange, 
and  leaking  will  ensue.  These  flanges  are  sometimes  fitted  with 
a  recess,  R  (Fig.  73  b),  into  which  a  strip  of  soft  metal,  such  as 


228  MACHINE    DESIGN 

copper,  can  be  caulked  to  check  small  leaks;  but  this  can  hardly 
be  considered  satisfactory  in  high-grade  work.  A  somewhat 
better  grip  of  the  flange  is  sometimes  obtained  by  rolling  the 
pipe  into  a  groove  in  the  flange,  as  shown  at  E,  Fig.  73  (b). 

In  the  so-called  Van  Stone  joint  (Fig.  73  c),  the  ends  of 
the  pipes  themselves  are  flanged  over  and  the  joint  made  between 
the  flanges  so  formed.  The  flanges  F,  F  become  clamps  for 
holding  the  flanges  proper  together,  and  may  be  loose  on  the  pipe. 
This  last  feature  is  a  very  useful  one,  as  it  greatly  facilitates 
erection.  This  form  of  joint  has  been  used  with  success  in  high- 
pressure  work. 

For  the  highest  grade  of  work,  wrought-steel  flanges  are 
welded  to  the  pipe,  making  the  pipe  and  its  flanges  one  piece. 
This  construction,  while  expensive,  is  almost  essential  for  large 
pipe  and  the  highest  pressures. 

To  prevent  the  packing  from  blowing  out,  the  flanges  are 
sometimes  fitted  with  a  recess  and  tongue  as  shown  at  H,  Fig.  73 
(b).  This  construction  is  essential  for  very  high  pressures,  as 
in  hydraulic  work,  but  should  be  avoided  if  possible  in  steam 
lines,  as  it  makes  it  difficult  to  renew  the  packing.* 

Although  several  efforts  have  been  made  to  establish  standard 
dimensions  for  pipe  flanges,  several  systems  are  in  common  use 
in  this  country.  The  most  important  of  these  systems  are  the 
standards  adopted  by  the  Flange  Standardization  Committee, 
of  the  A.  S.  M.  E.,  and  that  known  as  the  Manufacturers'  Standard. 
The  first  is  used  for  pressures  up  to  125  pounds  per  square  inch 
and  the  second  for  pressures  up  to  250  pounds  per  square  inch. 
The  student  is  referred  to  standard  handbooks,  and  the  catalogues 
of  various  manufacturers  for  details  of  these  several  systems. 
See  also  Transactions  A.  S.  M.  E.,  Vol.  XXI. 

THIN   PLATES 

88.  General  Theory.  The  theory  of  the  stresses  induced  in 
thin  plates,  when  subjected  to  load  pressures,  is  one  of  the  most 
uncertain  portions  of  the  mechanics  of  materials,  due  in  part  to 

*  For  a  fuller  description  of  Van  Stone   joints  see  articles  by  W.  F.  Fischer, 
Power,  Feb.  23,  1909,  and  March  2,  1909. 


TUBES,  PIPES,  CYLINDERS,  FLUES,  AND  THIN  PLATES       22Q 

the  complexity  of  the  problem,  and  in  part  to  the  scarcity  of 
corroborative  experimental  data.  The  subject  has  been  investi- 
gated mathematically  by  Grashof,  Bach,  Unwin,  Merriman,  and 
others,  and  the  experimental  work  of  Bach,  Benjamin,  and  others 
verifies  in  a  measure  some  of  their  conclusions.  The  mathematical 
results  obtained  by  various  authorities  differ  mainly  in  the  co- 
efficients, the  general  form  of  the  equations  being  in  most  cases 
similar.  Those  due  to  Merriman*  will  be  used  in  this  treatise 
as  they  are  simple,  easy  to  apply,  and  give  values  as  safe,  generally, 
as  any  others. 

Let  t  =  the  thickness  of  the  plate  in  inches;  r  =  the  radius 
of  circular  plate  in  inches;  /  =  length  and  b  =  breadth  of  rect- 
angular plate  in  inches;  p  =  maximum  tensile  stress;  w  =  load 
per  unit  area  in  Ibs. ;  and  P  =  concentrated  load  in  Ibs.  Then 

For  flat  circular  plates  supported  but  not  fixed  at  the  edges, 
carrying  a  distributed  load 

—  for  wrought  iron  or  steel        .      .      .      .      (i) 


and  t  =  r  \l  —  —  for  cast  iron (2) 

*  o   f 

For  flat  circular  plates  fixed  at  the  edges  (encastre),  carrying 
a  distributed  load 


2     W 

t  =  r  \ for  wrought  iron  or  steel     .      .      .      .      (3) 

1  3    P 

and  t  =  r  \l  —  — -  for  cast  iron       . (4) 

^  4    P 

For  flat  circular  plates  supported  but  not  fixed  at  the  edges, 
carrying  a  concentrated  load  P,  which  is  applied  to  the  centre 
of  the  plate  over  a  small  circle  of  radius  ro  so  that  P  =  TT  f2  wo 

t  =  ro  ^  Fi  +  2  lo  ge  -      — °  for  steel  or  wrought  iron  (5) 
and/  =  r0\  \  |  +  ^/0&,  - -~]  ~  for  cast  iron     .      .      .      (6) 

J    L-5  4  ^0  J    P 

*  See  Merriman's  "  Mechanics  of  Materials,"  1907  edition,  page  409. 


230  MACHINE    DESIGN 

For  flat  circular  plates  encastre  carrying  a  concentrated  load 

P,  which  is  applied  to  the  centre  of  the  plate  over  a  small  circle 
of  radius  ro. 

I  r~2          4  T  — i  W0 

t  =  ro  ^     — f-  —  lo  ge  —  — °  for  steel  or  wrought  iron  (7) 
LO        o  o  — 1  .r 

I  r"Z  T  — \  w 

and  t  =  r  ^     -  +  -  loge  —     — °  for  cast  iron   .      .      .      (8) 

In  equations  (5),  (6),  (7)  and  (8)  wo  is  the  pressure  per  unit 

P 

area  on  the  small  circle  whose  radius  is  r  ,  i.e.,  wo  =  — 2.      The 

-•'»  ^   ^ 

value  of  wo  should  not  exceed  the  elastic  strength  of  the  material. 
Example.  A  circular  cast-iron  plate  20"  in  diameter  supports 
a  load  of  4,000  Ibs.  at  its  centre,  the  load  being  applied  by  a  bolt 
whose  head  is  2"  in  diameter.  How  thick  must  the  plate  be,  if 
simply  supported,  in  order  that  the  tensile  strength  in  it  shall  not 
exceed  6,000  Ibs.  per  square  inch? 

Here  P  =  4,000;  r  =  10;  ro  =  i  and  p  =  6,000. 

.' .  w   = 2  =  -      — -  =  1,2615  Ibs. 

TT  r02       x  x  i2 

Y  IO 

and  I  o  ge  —  =  I  o  ge  —  =  2.3  whence  from  equation  (6) 


'•vii+V 

Rectangular  Plates.  If  2/  and  26  are  the  length  and  breadth 
of  a  rectangular  plate,  then  for  plates  supported  but  not  fixed, 
and  for  a  uniformly  distributed  load  w  per  square  inch 


.      •      •    .,     -     (9) 
*  *  \*   ~r  u  j  r 

and  for  fixed  edges 


TUBES,  PIPES,  CYLINDERS,  FLUES,  AND  THIN  PLATES      231 
If  /  =  b  equation  (9)  reduces  to 


and  equation  (10)  reduces  to 

'=Wjf (") 

The  above  equations,  as  before  stated,  are  not  to  be  relied  on 
implicitly,  but  will  serve  as  approximate  guides  only.  This  is 
particularly  true  in  cast  materials  where  heavy  ribbing  is  used 
and  where  trained  judgment  is  perhaps  the  best  guide. 

89.  Flat  Stayed  Surfaces.  One  of  the  most  important  cases 
of  flat  plates  occurs  in  boiler  work,  where  large  flat  areas  are  held 
against  pressure  by  stays  at  regular  intervals  over  the  surface. 
These  stays  are  usually  screwed  into  the  plate  and  the  projecting 
end  is  slightly  riveted  over  to  insure  steam  tightness.  The  various 
Inspection  Bureaus  and  Insurance  Companies  give  practical 
formulae  for  the  design  of  such  plates,  and  these  can  be  safely 
used.  Thus  the  U.  S.  Board  of  Supervising  Inspectors*  and  the 
American  Boiler  Makers'  Association  rules  give  for  steel  plates 

cxf 


Where  w  =  pressure  in  Ibs.  per  sq.  in.,  t  =  thickness  of  plate  in 
sixteenths  of  an  inch,  s  =  greatest  pitch  of  stays  in  inches,  and  C  — 
a  constant  as  below  given 

C  =  1 1 2  for  plates  ^"  thick  and  under. 
C  =  i2o  "        "        over  •&»  thick. 

C  =  140  "         "        with  stays  having  a  nut  inside  and  outside. 
C  =  i6o  "         "  "       "         "        washers  .5  as  thick  as 

the  plate  and  of  a  diameter  at  least  .5  the  greatest  pitch. 

*  See  General  Rules  and  Regulations  of  U.  S.  Supervising  Inspectors;   also 
Rules  of  American  Bureau  of  Shipping. 


CHAPTER  X 
CONSTRAINING  SURFACES 

go.  General  Considerations.  As  the  various  members  of  a 
machine  must  move  with  definite  relative  motion,  they  must  be 
retained  in  correct  position  by  constraining  surfaces.  Thus 
a  shaft  is  held  in  position  by  bearings  which  locate  its  axis 
of  rotation,  and  by  collars  which  prevent  motion  endwise.  The 
relative  motion  of  a  pair  of  constrained  members  may  be  that  of 
sliding,  as  in  the  case  of  an  engine  crosshead  and  its  guide; 
rotation,  as  in  the  case  of  a  shaft  journal  and  its  bearing;  rolling, 
as  in  roller  and  ball  bearings;  or  a  combination  of  some  of  these 
as  illustrated  in  certain  forms  of  cams,  where  both  sliding  and 
rolling  exist.  Dry  metallic  surfaces,  under  any  appreciable  load, 
even  when  smoothly  machined,  will  not  slide  over  each  other 
without  abrasion.  It  is  therefore  necessary  to  keep  rubbing 
surfaces  separated  by  a  thin  film  of  some  kind  of  lubricant,  and 
the  whole  subject  of  the  design  of  constraining  surfaces  is  closely 
connected  with  the  theory  of  lubrication.* 

It  has  been  pointed  out  in  Chapter  IV,  that  when  bath  or 
forced  lubrication  is  maintained,  the  friction  between  two  rubbing 
surfaces  is  independent  of  the  character  of  the  material  of  which 
the  surfaces  are  composed;  but  when  the  surfaces  are  "  imper- 
fectly" lubricated  the  frictional  resistance  depends  somewhat  on 
the  metals  used.  Experience  has  shown  that  like  metals  usually  do 
not  rub  together  well.  Thus  steel  on  steel  (except  when  hardened) , 
steel  on  wrought  iron,  or  cast  iron  on  cast  iron,  are  poor  combina- 
tions except  where  the  velocity  is  low  and  the  pressure  light.  If  two 
rubbing  surfaces  of  cast  iron  can  be  run  together  for  some  time 
without  cutting,  they  take  on  hard  glazed  surfaces  which  will  run 
well  together.  This  is  well  illustrated  in  slide  valves  and  pistons  of 

*  See  Chapter  IV. 

232 


CONSTRAINING    SURFACES  233 

steam  engines.  Care  must  be  exercised  that  the  surfaces  are  well 
lubricated  when  first  put  in  service.  Soft  steel  and  wrought  iron 
will  both  run  well  on  hardened  steel,  and  hardened  steel  may 
be  run  on  hardened  steel  at  very  high  pressures  and  velocities,  if 
the  surfaces  are  ground  true,  and  polished.  Steel  and  wrought 
iron  will  run  very  well  on  brass  or  bronze.  The  alloys  of  copper, 
tin,  zinc,  antimony,  lead,  etc.,  commonly  known  as  anti -friction 
or  babbitt  metals,  run  extremely  well  with  steel  or  wrought- 
iron  journals.*  Innumerable  alloys  of  this  kind  are  upon 
the  market  under  different  names.  They  can  be  made  of  any 
degree  of  hardness,  depending  largely  upon  the  proportion  of 
antimony  used.  Very  hard  alloys  of  this  kind  are  sometimes 
known  as  white  brass.  In  using  babbitt  metal  for  heavy  pres- 
sures, care  should  be  exercised  that  the  particular  alloy  selected 
is  hard  enough  so  as  not  to  flow  under  the  applied  pressure. 
Other  materials,  such  as  wood,  are  sometimes  used  for  rubbing 
surfaces.  The  conditions  which  influence  the  selection  of 
materials  for  rubbing  surfaces,  and  the  practical  considerations 
governing  their  application,  will  be  more  fully  discussed  in  con- 
nection with  the  several  forms  of  constraining  surfaces. 

The  most  common  forms  of  motion  in  machines  are  rectilinear 
translation  and  rotation;  therefore  the  most  important  forms  of 
constraining  surfaces  are 

(a)  Sliding    surfaces,    for    the    constrainment   of   rectilinear 

motion. 

(b)  Journals  and  bearings,  for  the  constrainment  of  motion 

of  rotation. 

SLIDING   SURFACES 

91.  Forms  of  Sliding  Pairs.  The  stationary  member  of  a  pair 
of  surfaces,  which  have  relative  sliding  motion,  is  usually  called 
the  guide,  while  the  moving  part  has  various  names  depending 
on  the  service,  as  the  ram  of  a  shaping  machine,  the  table  of  a 
planing  machine,  or  the  crosshead  of  an  engine.  The  general 
term  sliding  member  will  be  used  here  to  denote  the  moving 

*  See  Kent's  "  Mechanical  Engineer's  Pocket  Book  "  for  detailed  analysis  and 
properties  of  some  of  the  best  known  alloys. 


234  MACHINE    DESIGN 

member.  Sliding  pairs  may  be  classified  by  the  degree  of  lateral 
constrainment  afforded  the  slider  by  the  guides,  and  this  may  be 

(a)  Partial  lateral  constrainment. 

(b)  Complete  lateral  constrainment. 

In  either  case  the  rubbing  surfaces  of  the  guide  and  sliding 
member  may  be  either  square,  angular,  or  circular.  Thus  Fig. 
74  (a)  shows  a  form  of  angular  guide  much  used  on  planing  ma- 
chines while  Fig.  74  (b)  shows  a  set  of  square  guides  for  a  similar 
purpose.  In  each  case  the  lateral  constrainment  is  only  partial, 
the  tendency  of  the  platen  to  raise  being  resisted  by  gravity. 
Fig.  75  (a)  shows  the  crosshead  of  a  steam  engine  with  an  angular 
guide.  Here,  lateral  constrainment  is  complete.  Fig.  75  (b)  is 
also  a  steam-engine  crosshead  with  circular  guiding  surfaces. 
This  form  of  surface  may  be  considered  as  a  special  form  of  the 


FIG.  74  (a).  FIG.  74  (b). 

angular  type.  If  the  circular  guiding  surfaces  have  a  common 
centre  at  O,  the  crosshead  is  prevented  from  rotating  around  O 
only  by  the  connecting-rod ;  and  as  long  as  it  is  so  held  from 
rotating  the  lateral  constrainment  is  complete.  If  the  surfaces 
have  different  centres  as  Ol  O2,  it  is  obvious  that  rotation  cannot 
take  place.  Figs.  76  (a)  and  76  (b)  show  square  and  angular 
guides  where  constrainment  is  complete. 

The  characteristic  which  distinguishes  the  square  guide  from 
the  angular  one  is  that  in  the  square  guide  two  sets  of  adjustments 
must  be  made  to  compensate  for  wear,  while  in  the  case  of  the 
angular  guide  one  set  only  is  needed.  Thus  in  Fig.  76  (a),  vertical 
wear  must  be  compensated  for  by  lowering  the  piece  A,  while 
lateral  wear  is  taken  up  by  the  set  screws  C  which  press  against 
the  wearing  strip  or  gib  B.  In  Fig.  76  (b)  lateral  and  vertical  wear 
are  both  compensated  for  by  the  set  screws  C  which  press  upon 
the  gib  D.  Sometimes  D  is  made  tapering  and  provided  with  a 


CONSTRAINING    SURFACES 


235 


screw  adjustment  so  that  it  can  be  moved  endwise,  thus  com- 
pensating for  wear.  In  such  cases  the  set  screws  C  are  omitted. 
As  to  the  relative  merits  of  square  and  angular  guiding  sur- 
faces, it  may  be  said,  in  general,  that  square  surfaces  are  easier  to 
machine  and  fit  than  the  angular  ones.  There  are  many  places, 
however,  such  as  the  cross  slides  of  lathe  carriages,  where  the 
angular  guide  is  much  more  convenient.  In  places  such  as  lathe 
beds  the  V  guides  commonly  used  have  the  advantage  of  auto- 
matically taking  up  lost  motion,  no  matter  how  badly  they  are 
worn.  But,  as  a  rule,  the  bearing  surfaces  of  such  V  guides  are 
very  small  and  wear  soon  begins  to  be  apparent,  especially  as  the 
wear  from  the  carriage  is  usually  concentrated  on  a  short  por- 


(a) 


FIG.  75. 


(a) 


FIG.  76. 


tion  of  the  bed.  There  is  a  tendency  among  manufacturers  to 
discard  the  V  guide  in  favor  of  flat  surfaces.  English  practice, 
especially  in  large  tools,  is  in  advance  of  American  practice  in  this 
particular.  A  combination  of  V  and  flat  guides  is  also  often 
used. 

92.  General  Principles.  If  a  short  block,  Fig.  77,  slides 
backward  and  forward  upon  another  member  B,  carrying  a  fixed 
load  P,  it  is  evident  that,  if  the  material  in  A  and  B  were  homo- 
geneous and  the  velocity  were  uniform  throughout  the  stroke, 
the  frictional  resistance  and  consequent  wear  would  be  practically 
uniform  over  the  whole  surface  of  B.  These  conditions  are 
difficult  to  attain  and  seldom  occur  in  practice.  Since  A  must 
be  stopped  and  started  at  each  end  of  the  stroke,  it  follows  that 
the  velocity  cannot  be  uniform;  although  in  some  machines  such 
as  plate  planers  this  condition  is  approximated.  Usually,  how- 


236  MACHINE    DESIGN 

ever,  the  velocity  varies  from  zero  at  the  beginning  to  a  maximum 
somewhere  near  the  middle  of  the  stroke,  as  in  the  case  of  engine 
crossheads,  shaping  machines,  etc.  Again,  the  load  P  may  vary 
greatly.  Thus  in  the  steam  engine,  the  normal  pressure  P  be- 
tween the  crosshead  and  its  guide  is  zero  at  each  end  of  the  stroke 
and  a  maximum  near  mid-stroke.  The  velocity  of  the  crosshead 
also  varies  from  zero  at  each  end  of  the  stroke  to  a  maximum 
near  mid-stroke.  In  the  ordinary  case  the  greatest  frictional 
resistance*  and  wear  will  occur  near  mid-stroke,  because  both 
velocity  and  normal  pressure  Between  the  bearing  surfaces  are 
greatest  at  this  position.  If  the  crosshead  could  be  made  the 
same  length  as  the  guide,  the  unit  bearing  pressure,  at  the  middle 
of  the  stroke,  would  be  practically  uniform  over  the  whole  surface, 
and  would  be  small  compared  to  the  unit  normal  pressure  attained 
when  the  crosshead  is  short.  For  positions  of  the  crosshead  near 
mid-stroke  the  wear  would  be  approximately  equal  over  the  whole 
surface,  and  much  less  than  when  the  crosshead  is  very  short, 
but  still  theoretically  greater  than  at  the  end  positions,  when  both 
velocity  and  normal  pressure  are  zero.  It  has  been  found  by 
experience  that  when  the  sliding  block  and  guide  are  made 
the  same  length,  the  wear,  even  under  varying  load  and  ve- 
locity, is  very  small,  and  more  uniform  over  the  entire  contact 
surfaces. 

It  is  seldom  possible,  however,  to  make  the  sliding  member 
the  same  length  as  the  guide.  Thus  in  lathe  carriages,  the  rams 
of  shaping  machines,  and  the  tables  of  planing  machines,  the 
sliding  member  is,  in  some  machines,  shorter  than  the  guide,  and 
in  other  machines  longer.  In  most  cases  of  this  kind  the  wear 
is  liable  to  be  greater  on  one  part  of  the  guide,  or  sliding  member, 
than  on  another.  Thus  in  the  case  of  a  shaping  machine  the 
ram  seldom  operates  at  full  stroke,  and  the  wear  on  the  back  end 
of  the  ram  is  very  small,  the  result  being  that  when  appreciable 
wear  takes  place  on  the  forward  end  of  the  ram,  and  the  guides 
are  readjusted  to  compensate  for  the  same,  the  back  end  of  the 
ram  will  not  pass  through  the  guides  at  all,  hence  the 

*  See  Art.  32. 


CONSTRAINING    SURFACES 


237 


adjustment  must  be  somewhat  slack,  and  accurate  work 
cannot  be  done.  In  other  machines  the  excessive  local  wear 
comes  on  the  guide,  and  a  similar  result  occurs.  Professor  Sweet* 
has  corrected  this  difficulty,  in  certain  machines  which  he  has 
built,  by  reducing  the  wearing  surface  on  that  portion  of  the  slid- 
ing member  or  guide,  as  the  case  may  be,  where  the  tendency  to 
wear  is  least.  He  has  suggested  the  following  convenient  method 
of  laying  out  the  wearing  strips  on  the  surface  of  a  sliding  member. 
Fig.  78  shows  a  sliding  surface  such  as  is  found  on  the  ram  of  a 
shaping  machine,  where  little  wear  occurs  on  the  back  or  right- 
hand  end  as  here  shown.  The  shaded  portions  represent  the 
parts  of  the  surface  which  have  been  relieved,  leaving  the  wearing 


FIG.  77. 


FIG.  78. 


FIG.  79. 


strips  5,  S{  and  S2,  etc.  To  lay  off  the  surface,  draw  the  diagonal 
a,  b  across  the  surface  to  be  relieved.  From  a  draw  the  line  ac, 
making  any  convenient  angle  with  the  horizontal.  Lay  off  ce 
equal  to  the  width  of  the  face  x.  Draw  de  parallel  to  ac  and  take 
the  vertical  distance  above  the  point  of  intersection  of  ab  and  de 
for  the  first  gap,  and  the  corresponding  vertical  distance  below 
the  point  of  intersection  for  the  first  wearing  strip,  repeating  this 
operation  to  the  end  of  the  surface.  Similar  wearing  strips  should 
be  cut  in  the  opposite  direction  on  the  other  member,  if  it  is 
comparatively  long;  but  where  a  short  block  slides  in  a  long 
guide,  the  guide  only  need  be  relieved. 

93.  Bearing  Pressures  on  Sliding  Surfaces.  It  is  pointed  out 
in  Article  32  that  the  tendency  of  a  loaded  flat  surface  to  expel 
the  lubricant  is  resisted  to  a  certain  degree  by  the  viscosity  of 


*  Professor  Sweet  has  embodied  some  of  his  experience  in  this  line  in  a  little 
book  called  "Things  that  are  Usually  Wrong,"  which  will  well  repay  reading. 


238  MACHINE    DESIGN 

the  lubricant,  and  its  power  to  adhere  to  the  stationary  member. 
This  resisting  power  is  much  less  marked  in  sliding  surfaces  than 
in 'rotating  surfaces,  as  here  the  motion  is  intermittent.  It  is 
difficult  therefore  to  lubricate  sliding  surfaces  as  efficiently  as 
rotating  surfaces,  and,  in  general,  they  must  be  considered  as 
"imperfectly"  lubricated  surfaces.  The  unit  bearing  pressure 
that  can  be  sustained  by  sliding  surfaces  is,  therefore,  much  less 
than  can  be  borne  by  rotating  journals.  Further,  it  is  difficult  to 
obtain  initially  true  sliding  surfaces  and,  as  pointed  out  above, 
very  difficult  to  maintain  their  accuracy  under  service.  The 
sliding  part,  and  also  the  guides  themselves,  should,  therefore,  be 
designed  for  rigidity;  in  fact  considerations  of  strength  seldom 
need  to  be  considered,  but  the  guides  should  be  so  stiff  that 
localized  pressure  will  not  occur.  It  is  not  surprising,  in  view 
of  these  considerations,  that  the  allowable  bearing  pressures  as 
fixed  by  practice  vary  greatly,  even  with  similar  classes  of  work. 
Owing  to  the  difficulties  of  lubrication  and  compensation  for  wear, 
it  may  be  stated,  as  a  general  principle,  that  the  bearing  pressure 
must  be  kept  so  low  that  wear  is  inappreciable,  if  accurate  sur- 
faces are  to  be  maintained. 

The  following  are  average  values  of  bearing  pressures  for 
different  forms  of  sliding  surfaces,  as  fixed  by  practice: 

Crossheads,*  stationary  slow-speed  engines  30  Ibs.  to  50  Ibs. 

high       "          "       10  "     "  30  " 
"  marine  engines  50  "     "  75  " 

94.  Lubrication  of  Sliding  Surfaces.  Sliding  surfaces  are  very 
difficult  to  lubricate  efficiently  on  account  of  the  "wiping"  action 
of  the  sliding  member.  In  high-speed  engines,  bath  lubrication 
is  commonly  obtained  by  enclosing  the  running  parts,  and  allow- 
ing them  to  run  in  what  practically  amounts  to  an  oil  bath. 

Where  this  cannot  be  done,  care  must  be  exercised  in  the 
manner  in  which  the  lubricant  is  supplied.  If  possible,  when 
the  guide  is  horizontal,  the  lubricant  should  be  supplied  near 
the  middle  of  the  guide.  The  oil  grooves  in  the  moving  part  should 
also  be  given  careful  consideration.  From  the  theory  of  lubrica- 

*  See  Trans.  A.  S.  M.  E.,  Vol.  XVIII,  page  753. 


CONSTRAINING    SURFACES  239 

tion  it  is  evident  that  the  oil  channels  on  all  constraining  surfaces 
should  be  at  right  angles  to  the  direction  of  motion,  wherever  the 
velocity  is  great  enough  to  draw  lubricant  between  the  surfaces. 
If  made  otherwise  their  effect  is  to  relieve  any  tendency  to  form 
a  pressure  film.  The  grooves  in  crossheads,  and  other  sliding 
members,  should,  therefore,  be  made  as  in  Fig.  79  (a)  and  not  as 
in  79  (b).  In  either  case  the  grooves  should  be  stopped  some 
distance  from  the  edge  of  the  surface  so  as  not  to  facilitate  the 
escape  of  the  oil.  When  the  load  is  so  heavy  that  forced  lubrica- 
tion must  be  used,  the  system  of  grooves  shown  in  Fig.  79  (b)  is 
correct;  the  oil  being  forced  in  at  O.  Care  should  also  be  taken 
that  the  outer  edges  of  the  slider,  and  the  edges  of  the  oil  grooves, 
are  chamfered  so  as  to  assist  the  entrance  of  the  lubricant.  If 
the  edges  are  square  and  sharp  their  scraping  effect  may  seriously 
impair  the  lubrication.  Where  the  guiding  surfaces  are  very 
long,  as  in  planing  machines,  oiling  devices  such  as  rollers  dipping 
in  an  oil  pocket,  placed  at  intervals  along  the  guides,  are  very 
effective. 

JOURNALS  AND  BEARINGS 

BEARINGS 

95.  Forms  of  Bearings.  The  part  of  a  machine  frame,  or 
other  member,  which  constrains  a  rotating  member,  such 
as  a  shaft,  is  known  as  a  bearing.  That  portion  of  the 
rotating  member  which  engages  with  the  bearing  is  known  as  a 
journal.  Journals  are  necessarily  circular  in  all  cross-sections, 
but  their  profile  may  be  cylindrical,  conical,  spherical,  or  even 
more  complex  in  form,  as  in  the  case  of  thrust  bearings.  (See 
Art.  105.) 

One  or  more  of  the  following  considerations  affect  the  design 
of  the  bearing  proper:— 

(a)  Rigidity,  in  order  that  the  alignment  may  not  be  seriously 

affected  by  deflection. 

(b)  Strength,  to  resist  rupture  under  the  greatest  loads. 

(c)  Adjustment,  to  compensate  for  wear. 

(d)  Formation  and  maintenance  of  an  oil  film. 

(e)  Automatic  adjustment,  to  insure  alignment. 


240  MACHINE    DESIGN 

(a  and  b) .  The  inside  diameter,  or  bore  of  the  bearing,  and  also 
its  length  are  fixed  by  the  dimensions  of  the  journal  which  engages 
with  it;  and  the  required  strength  and  rigidity  may  be  secured 
by  a  proper  distribution  of  metal  in  accordance  with  the  general 
principles  discussed  in  Chapter  III,  and  which  apply  to  all  forms 
of  bearings,  as  far  as  strength  and  stiffness  are  concerned. 

Usually  the  question  of  strength  does  not  enter  into  the  design 
of  the  main  part  of  the  bearing.  If,  however,  the  cap  A,  Fig.  80, 
should  be  called  upon  to  carry  the  load,  as  is  often  the  case,  its  di- 
mensions should,  in  general,  be  checked  for  strength,  and  its  design 
should  be  such  that  stiffness  is  secured.  The  exact  distribution  of 
the  pressure  over  a  bearing  is  not  known;  but  the  assumption 
that  the  cap  is  a  beam  loaded  at  the  centre  and  of  a  length  equal 
to  the  distance  between  the  cap  bolts  will  give  dimensions  on  the 
safe  side  for  strength  and  deflection.  The  greatest  bending 
moment  and  deflection  for  such  beams  are  given  in  Case  IX, 
Table  i.  It  is  impossible  to  adjust  the  cap  bolts  so  as  to  be  sure 
that  the  load  is  uniformly  distributed  among  them,  and  the  un- 
certainty of  the  initial  stress  due  to  screwing  up  the  nuts  makes 
the  problem  more  difficult.  For  this  reason  the  cap  bolts  should 
be  designed  to  carry  more  than  the  apparent  load.  If  only  two 
bolts  are  used  each  should  be  designed  for  two-thirds  of  the  total 
load;  if  four  are  used  each  should  be  able  to  carry  one-third  of 
the  load  with  an  apparent  stress  of  not  more  than  6,000  Ibs.  per 
square  inch. 

The  last  three  items,  c,  d  and  e,  affect  the  form  of  the  bearing. 
Consider  first  c  and  d.  It  is  evident  that  the  metal  of  the  bearing 
will  wear  away  most  rapidly  in  the  line  of  greatest  pressure,  hence 
adjustment  for  wear  should  also  be  along  this  line.  It  follows 
also  that  the  bearing  should  be  parted  at  right  angles  to  the  line 
of  greatest  pressure.  Thus,  if  the  load  on  the  shaft  be  a  simple 
vertical  load  P,  as  in  Fig.  80,  wear  will  take  place  only  on  the 
bottom  half  of  the  bearing.  If  this  wear  is  so  small  as  not  to 
interfere  with  the  alignment  of  the  shaft,  or  if  all  the  bearings 
on  the  shaft  wear  uniformly,  adjustment  may  be  made  by  lowering 
the  cap  A.  If  the  shaft  must  occupy  a  fixed  position  relative  to 
the  frame  of  the  machine,  alignment  must  be  maintained  by 


CONSTRAINING    SURFACES 


241 


raising  the  lower  bearing  surface.  Where  this  is  desirable  the 
lower  wearing  surface  is  usually  made  separate  from  the  pillow- 
block,  as  in  Fig.  82,  thus  allowing  the  bearing  to  remain  fixed 
in  position,  while  the  wearing  part  may  be  raised  to  compensate 
for  the  wear.  If  the  load  P,  Fig.  80,  be  in  an  upward  direction, 
all  necessary  adjustment  may  be  made  by  means  of  the  cap. 

It  was  shown  in  Articles  32  and  33  that  a  journal  will  auto- 
matically tend  to  form  a  film  of  lubricant  between  itself  and  the 
bearing.  If  the  conditions  under  which  the  lubricant  is  supplied 
are  correct,  fluid  pressure  may  thus  be  created  between  the 
journal  and  bearing  provided  the  surface  of  the  bearing  is  con- 
tinuous for  some  distance  on  each  side  of  the  line  of  action  of  the 


FIG.  80. 

load.  The  greatest  pressure  will  be  found  near  this  line  of  action. 
It  is  evident  that  the  bearing  shown  in  Fig.  80  fulfills  both  these 
requirements  for  vertical  load  either  upward  or  downward;  but 
is  unsuited  for  lateral  pressure  from  the  standpoints  both  of  ad- 
justment for  wear  and  lubrication. 

Suppose,  however,  that  the  journal  carries  a  heavy  vertical 
load  P  (Fig.  81),  and  is  subjected  at  the  same  time  to  a  heavy 
horizontal  belt  pull  Pt.  The  resultant  of  these  forces  is  P2,  and 
the  arrangement  of  parts  shown  in  Fig.  81  is  correct  for  motion 
of  rotation  in  either  direction.  If  Pl  be  reversed  in  direction 
the  resultant  of  Pj  and  P2  will  be  P3,  and  the  arrangement  is  not 
correct  for  adjustment  against  wear,  and  very  defective  as  far  as 
lubrication  is  concerned,  as  the  surface  is  broken  near  the  point 

16 


242 


MACHINE    DESIGN 


where  the  greatest  film  pressure  should  exist.  Bearings  of  this 
form  are  often  used  in  steam-engine  work,  and  in  such  cases 
the  force  Pv  due  to  the  steam  pressure  on  the  piston,  is 
continually  reversed  in  direction.  Another  adjustment  for 
a  similar  case  is  shown  in  Fig.  82.  Here  the  shoe  or  bottom 
"brass"  can  be  raised  up  by  introducing  thin  "  shims,"  or  liners, 
underneath  it;  while  lateral  wear  can  be  taken  up  by  setting  out 
the  " cheek  pieces"  #,  by  means  of  the  wedges  D.  Provision  is 
thus  made,  by  this  arrangement,  for  taking  up  wear  in  all  direc- 
tions and  keeping  the  shaft  accurately  aligned  and  located.  For 
horizontal  pressures  in  either  direction  the  resultant  P3  passes 
close  to  the  point  at  which  the  bearing  is  parted;  and  hence  the 
best  conditions  for  lubrication  do  not  exist.  Pressure  films  more 


FIG.  83. 


or  less  perfect,  depending  on  the  oil  supply,  will  form  on  the  lower 
shoe,  but  the  continual  reversing  of  the  lateral  pressure  P,  hardly 
allows  time  for  the  formation  of  pressure  films  on  the  cheeks. 
These  reversals  in  pressure,  however,  allow  the  lubricant  to  be 
carried  by  the  shaft,  first  under  one  cheek,  and  then  under  the 
other,  thus  lubricating  them  effectively. 

Sometimes  a  bearing  consists  of  a  conical  bushing  split  at 
some  convenient  place,  as  shown  in  Fig.  83.  By  releasing  the 
nut  A,  and  screwing  up  on  B,  the  bushing  may  be  forced  into 
the  frame  C,  thus  closing  the  bore  of  the  bushing  slightly  and 
compensating  for  wear.  It  is  obvious  that  once  the  bore  of  the 
bushing  is  worn  eccentric,  no  amount  of  taking  up  can  rectify 
its  shape;  in  fact  taking  up  wear  in  this  manner  tends  to  destroy 
the  fit  of  the  journal  in  the  bearing.  Occasionally  the  journal 


CONSTRAINING    SURFACES 


243 


itself  is  made  conical,  and  adjustment  for  wear  is  made  by  moving 
ttie  shaft  endwise.  The  application  of  such  bearings  is  limited 
to  short  shafts,  such  as  machine-tool  spindles. 

Machine  bearings  are  made  in  many  forms,  depending  on 
the  location  and  service.  The  bearings  are  sometimes  split 
into  three  pieces,  and  various  other  means  of  compensating  for 
wear  are  used,  but  the  fundamental  principles  outlined  above 
regarding  the^point  where  the  bearing  should  be  parted  apply 
to  all  forms. 

Consider  the  last  item  (e,  automatic  adjustment).  In  long 
lines  of  shafting,  which  tend  rapidly  to  get  out  of  adjustment,  it 
is  desirable  that  the  bearing  be  so  constructed  as  to  adjust  itself 
automatically  to  the  chang- 
ing position  of  the  shaft, 
in  order  to  avoid  localized 
pressure,  which  would  re- 
sult in  heating.  In  fast- 
running  machinery,  also, 
such  as  countershafts,  dy- 
namos, and  motors,  where 
perfect  alignment  is  neces- 
sary, self-adjusting  bearings 


FIG.  84. 


have  been  found  almost 
essential.  Fig.  84  shows  a 
bearing  of  this  kind  as  used  in  dynamo  and  motor  bearings. 
The  sleeve  A  has  a  spherical  surface  turned  upon  the  outside, 
the  centre  of  the  surface  being  at  O.  This  surface  engages  with 
a  similar  surface  bored  in  the  outer  casing  B.  The  sleeve  may 
swivel  in  any  direction,  but  the  centre  line  of  the  shaft  must 
always  pass  through  O.  When  a  shaft  has  only  two  bearings  of 
this  kind  it  is  evident  that  perfect  alignment  can  be  secured, 
within  the  range  of  motion  of  the  sleeves.  Similar  devices  are 
used  in  the  case  of  long  shafting,  where  many  bearings  must  be 
used.  It  is  obvious  that  the  fundamental  principles  regarding 
adjustment  for  wear  and  maintenance  of  the  oil  film,  apply  to  all 
bearings  of  this  form  also. 

96.  Practical   Construction  of  Bearings.     It   was   shown   in 


244  MACHINE    DESIGN 

Article  88  that  metal  such  as  brass,  bronze,  and  the  white  alloys 
make  excellent  bearing  surfaces  for  wrought-iron  or  steel  journals, 
on  account  of  their  an ti- friction  qualities.     It  is  to  be  noted  that 
even  in  the  case  of  perfect  lubrication,  where  the  character  of  the 
rubbing  surfaces  is  less  important  once  the  oil  film  is  established, 
care  must  be  exercised  in  the  selection  of  the  material  for  the 
bearing  surface,  in  order  that  abrasion  may  not  occur  before  the 
film  is  formed,  or  in  case  of  failure  of  the  film.     There  is  a  further 
advantage  in  having  the  bearing  surface  softer  than  the  journal, 
in  that  it  is  very  desirable  to  have  the  journal  maintain  its  form 
against  wear,  which  it  is  more  likely  to  do  when  rubbing  against 
a  soft  surface  than  it  would  against  one  harder  than  itself.     The 
bearing  itself  should  be  rigid,  so  as  to  insure  proper  alignment  of 
the  shaft.     Rigidity,  against  even  moderate  pressure,  could  not 
ordinarily  be  attained  if  the  entire  bearing  member  were  made 
of  the  white  alloys,  and  economy  prohibits  the  use  of  brass  and 
bronze  for  the  entire  bearing.     It  is  customary,  therefore,   to 
make  the  main  body  of  the  bearing  of  cast  iron  (or  sometimes  a 
steel  casting),  and  to  fit  into  it  wearing  surfaces  of  the  softer 
metals.     These  wearing  surfaces  may  be  either  rigidly  attached 
to  the  main  castings  or  may  be  removable.     In  Fig.  80  is  shown  a 
bearing  of  the  type  commonly  used  for  heavy  shafts  when  the 
babbitt-metal  lining  is  rigidly  attached  by  means  of  dovetail 
shaped  recesses,  into  which  the  babbitt  is  poured  in  a  molten 
state.     The  necessary  shrinkage  due  to  cooling,  which  would 
leave  the  lining  loose  in  the  recesses,  is  usually  overcome  by 
hammering  the  babbitt,  when  cold,  till  it  again  fills  the  recesses, 
and  then  boring  the  babbitt  to  size.     For  cheap  work  the  lining 
is  often  cast  to  size  on  a  metal  mandrel  and  no  further  work  put 
upon  it,  but  for  all  good  work  the  bore  of  the  lining  is  cast  small 
enough  to  allow  of  hammering  or  peening,  and  then  boring  to  a 
smooth  surface.     Fig.  81  shows  removable  linings  of  brass  or 
bronze  which  are  circular  in  section,  and  are  prevented  from 
turning  when  in  place  by  the  parting  pieced.      This -parting 
piece,  or   "  liner,"   also  permits  taking  up    wear    by    reducing 
its   thickness    as   occasion   requires.       Fig.    82    shows    an    ar- 
rangement of  wearing  surfaces  common  on  horizontal  steam- 


CONSTRAINING    SURFACES 


245 


engine  bearings.  The  cap  C  is  babbitted  with  some  form  of 
cheap  metal  since  there  is  no  wear  upon  it,  all  the  pressure  being 
either  downward  of  side  wise.  The  "quarter  boxes"  B,  and  the 
lower  box  or  shoe  A,  may  be  of  brass  or  bronze,  or  of  cast  iron 
lined  with  babbitt.  Where  there  is  danger  of  the  boxes  breaking, 
through  pounding  by  the  shaft,  and  where  it  is  desired  to  use  a 
babbitt  metal,  they  may  be  made  of  brass  or  bronze  and  babbitt- 
lined.  When  cast-iron  wearing  surfaces  are  used,  and  compensa- 
tion for  wear  is  important,  as  in  the  case  of  machine  tools,  it  is 
customary  to  make  the  wearing  surfaces  removable  as  indicated 
in  Fig.  81.  For  less  accurate  work  the  bearing  surface  is  part  of 
the  main  casting  itself,  machined  to  the  required  size.  Hardened 
steel  bearing  surfaces  are  obtained  by  making  circular  shells  or 
"bushings,"  of  the  required  internal  diameter,  and  of  sufficient 
thickness  to  insure  strength.  These  bushings  are  forced  into 
openings  in  the  main  casting  and  no  provision  for  taking  up  wear 
is  made.  If  the  forcing  operation  closes  the  bore  of  the  bushing, 
it  is  "lapped"  out  with  emery  and  oil  to  the  required  size.  Where 
the  bearing  must  work  under  water,  as  in  the  case  of  a  propeller 
shaft  or  the  lower  bearing  of  a  vertical  turbine  water  wheel,  a 
lining  of  lignum  vitae  or  other  hard  wood  is  often  used.  The 
surrounding  water  furnishes  the  only  lubricant  necessary  in  such 
cases.  A  detailed  description  of  the  many  arrangements  of 
bearing  surfaces  is  beyond  the  scope  of  this  treatise. 

When  the  bearing  must  work  under  trying  conditions,  as  on 
shipboard  or  in  a  heated  room,  and  there  is  some  question  as  to 
whether  the  heat  of  friction  will  be  dissipated  by  radiation,  the 
bearing  is  cast  hollow  so  that  water  may  be  circulated  around  it, 
thus  carrying  off  the  heat  and  maintaining  the  lubrication.  In 
an  emergency,  water  may  be  allowed  to  run  over  the  outside 
of  the  bearing,  accomplishing  the  same  purpose.  High-grade 
marine  work,  and  large  stationary-engine  installations,  are  often 
equipped  with  a  complete  system  of  water  circulation  on  the  most 
important  bearings. 

JOURNALS 

97.  Theoretical  Design  of  Journals.  The  considerations 
affecting  the  design  of  any  journal  are  one  or  more  of  the  following: 


246  MACHINE    DESIGN 

(a)  Strength  to  resist  rupture. 

(b)  Rigidity,  or  stiffness,  to  prevent  undue  yielding. 

(c)  Maintenance  of  form  against  wear. 

(d)  Maintenance  of  lubrication. 

(e)  Radiation  of  the  heat  due  to  frictional  resistance. 

The  first  two  considerations,  strength  and  rigidity,  are 
covered  by  the  general  principles  laid  down  in  Chapter  III,  and 
are  more  fully  considered  in  Chapter  XI,  where  the  special  prob- 
lems in  connection  with  shafts  are  discussed.  Economy  of 
material  dictates  that  iheminimum  diameter  of  shaft  be  consistent 
with  the  applied  bending  and  twisting  moments. 

The  third  consideration  (c)  particularly  affects  such  journals 
as  those  on  the  spindles  of  grinding  machines  and  machine  tools 
generally,  where  the  accuracy  of  the  product  depends  on  the  accu- 
racy of  the  journals.  Usually,  in  such  cases,  the  wearing  surface 
must  be  so  great,  in  order  to  reduce  the  wear  to  an  inappreciable 
amount,  that  the  consideration  of  strength  does  not  enter  into  the 
computations. 

The  considerations,  (d)  and  (e),  are  closely  correlated.  It 
was  shown  in  Articles  32  and  33  that  if  the  unit  bearing  pressure 
on  the  journal  is  not  too  great,  the  lubricant,  because  of  its 
viscosity,  may  be  drawn  in  between  the  journal  and  the  bearing, 
thereby  reducing  the  frictional  resistance.  This  frictional  re- 
sistance can  never  be  reduced  to  zero  even  with  perfect  lubrica- 
tion. The  energy  thus  absorbed  appears  as  heat,  and  is  radiated 
to  the  surrounding  air  by  the  metallic  surfaces  of  the  bearing, 
the  temperature  of  which  rises  till  the  rate  of  radiation  equals 
that  at  which  heat  is  being  generated.  In  well-designed  ma- 
chinery the  temperature  of  the  bearing  should  not  exceed  150°  F. 
The  raising  of  the  temperature  of  the  bearing  has  a  tendency  to 
lower  the  viscosity  of  the  lubricant,  and  if  the  bearing  becomes 
too  hot,  the  lubricant  becomes  so  thin  that  the  pressure  squeezes 
it  out  completely,  and  failure  of  the  bearing  by  abrasion  occurs. 
It  is  evident,  therefore,  that  a  journal  of  given  dimensions  may 
carry  a  given  load  very  satisfactorily  under  certain  conditions,  and 
fail  absolutely  under  others,  the  same  lubricant  being  used  in 
each  case.  The  consideration  of  the  proper  radiation  of  the  heat 


CONSTRAINING    SURFACES 


247 


generated  is,  therefore,  most  important.  It  may  be  assumed,  without 
serious  error,  that  the  rate  of  radiation  of  heat  is  proportional  to 
the  projected  area  of  the  bearing.  The  number  of  heat  units 
which  will  be  radiated  from  a  unit  of  surface,  at  any  given 
difference  in  temperature  between  the  bearing  and  the  surround- 
ing air,  is  a  fixed  quantity  for  any  set  of  conditions;  and  if  the 
heat  of  friction  per  unit  area  is-  greater  than  can  be  radiated 
at  the  desired  bearing  temperature,  the  temperature  of  the  bearing 
must  rise  till  equilibrium  is  obtained.  It  follows  therefore  that 
for  any  desired  bearing  temperature  the  work  of  friction  per  unit 
of  projected  area  of  bearing  must  not  exceed  the  rate  of  radiation 
per  unit  of  projected  area,  or 

TT 

P.  w  V  =  K     or     wV  =  -  (i) 

V- 

where  /*  is  the  coefficient  of  friction,  w  the  load  in  pounds  per 
unit  of  projected  area,  V  the  velocity  of  rubbing  in  feet  per 
minute,  and  K  the  rate  of  radiation  per  unit  of  projected  area  in 
foot  pounds  per  minute,  to  be  determined  experimentally. 

It  is  to  be  especially  noted  that  if  //  be  considered  as  constant, 
increasing  the  diameter  of  a  journal  (the  number  of  revolutions 
and  the  total  load  remaining  constant)  does  not  materially  affect 
the  development  or  dissipation  of  heat,  since  the  velocity  of  rub- 
bing is  increased  in  the  same  ratio  as  radiating  surface  is  in- 
creased. If,  however,  the  bearing  be  lengthened,  the  radiating 
surface  is  increased  and  the  work  of  friction  remains  unchanged, 
with  the  same  total  load  as  before.  This  last  statement,  while 
true  for  imperfectly  lubricated  surfaces,  is  only  approximately 
true  for  bearings  with  perfect  lubrication  as  will  be  seen 
presently. 

The  amount  of  heat  which  will  be  radiated  from  a  bearing 
has  been  experimentally  determined  by  Lasche.*  The  curves 
shown  in  Fig.  84  (a)  are  those  shown  in  his  Fig.  57,  transformed 
into  English  units,  and  with  the  scale  of  radiation  further  modified 
so  as  to  read  in  foot  pounds  per  square  inch  of  projected  area 
per  second,  instead  of  per  square  inch  of  actual  bearing  surface. 

*  See  Traction  and  Transmission,  January,  1903,  page  52. 


248 


MACHINE    DESIGN 


Curve  i  represents  actual  experimental  results,  with  bearings  of 
the  usual  proportions,  in  still  air.  Curve  2  is  for  bearings  which 
are  connected  to  large  iron  masses,  or  which  are  ventilated  by  air 
currents.  Curve  3  was  calculated  from  theoretical  considera- 


180 


160 


140 


*SV 


?  120 


W 


100 


80 


20 


0      2      4      6      8      10      12      14      16 

-li=RADIATION   IN  FT.  LBS.  PER  SEC.  PER  SQ.  IN.  PROJ.  AREA. 
60 

FIG.  84  (a). 

tions.  It  gives  the  radiation  from  a  very  thin  bearing  or  sleeve 
and  indicates  that  radiation  is  more  effective  as  the  bearing  be- 
comes thicker,  as  might  be  expected;  for  metal  is  a  better  con- 
ductor of  heat  than  air,  and  hence  the  thick  bearing  more  easily 


CONSTRAINING    SURFACES  249 

carries  the  heat  away  to  a  greater  radiating  surface.  The  values 
obtained  from  these  curves  may  therefore  be  used  for  K  in 
equation  (i).  Lasche  points  out  that  though  these  experiments 
represent  only  a  limited  variety  of  conditions,  they  are  probably 
on  the  safe  side  and  will  serve  at  least  as  a  very  useful  check  in 
designing. 

If,  in  designing  a  journal,  the  value  of  t*.  can  be  deter- 
mined, equation  (i)  and  Fig.  84  (a)  give  the  relations  which 
must  'exist  between  the  velocity  and  pressure  in  order  that  the 
safe  bearing  temperature  may  not  be  exceeded ;  or  if  the  pressure 
and  velocity  are  fixed  by  other  circumstances,  Fig.  84  (a)  indicates 
whether  radiation  must  be  assisted  by  artificial  means,  such  as 
water  circulation  or  currents  of  air. 

98.  Imperfectly  Lubricated  Journals.  It  has  been  shown  in 
Articles  32  and  33  that  the  value  of  /*,  for  imperfectly  lubricated 
surfaces,  is  a  very  variable  quantity,  even  for  the  same  simulta- 
neous values  of  velocity  and  pressure.  Not  only  does  it  vary  with 
velocity,  pressure,  and  temperature,  but  the  regularity  of  the 
oil  supply  (over  which  the  designer  has  little  control)  affects  it 
much  more  seriously.  Further,  bearings  running  under  the 
same  nominal  load  and  velocity  give  widely  different  values  of 
frictional  resistance  and  temperature  rise,  depending  on  whether 
the  load  is  constant  or  intermittent,  or  whether  the  motion  is 
steady  or  vibratory,  etc.  Notwithstanding  this,  equation  (i)  may 
be  made  to  serve  as  a  useful  check  in  doubtful  cases  by  assum- 
ing a  safe  value  of  M. 

The  assumption  is  sometimes  made  that  P.  is  a  constant ;  and 

TP 

formulae  of  the  form  w  V  =  —  =  C,  where  C  is  a  constant  that 

has  been  determined  from  practice,  are  much  used.  Thus  if  w  be 
expressed  in  pounds  per  square  inch  of  projected  area  and  V  in 
feet  per  minute,  Mr.  Fred  W.  Taylor*  gives  for  mill  work  C  = 
24,000;  and  says  that  C  =  12,000  is  not  safe  for  cast-iron  bearings 
with  ordinary  lubrication.  If  the  rise  of  temperature  in  the  bearing 
be  taken  as  75°  and  /*  be  taken  as  .015,  which  is  ordinarily  a  safe 

*  Transactions  A.  S.  M.  E.,  Vol.  XXVII. 


250  MACHINE    DESIGN 

TT 

value,  then  from  curve  i,  Fig.  84  (a),  K  =  222,  whence  C  =  —  = 

222 

-  =  15,000.  From  curve  2,  K  =  384  whence  for  ventilated 
.015 

bearings  C  =  -     -  =  25,600.       These   values  agree  with   Mr. 

Taylor's  limits  better  than  would  be  expected. 

All  formulae  of  this  empirical  form  must  be  considered,  as 
far  as  imperfectly  lubricated  journals  are  concerned,  as  applying 
only  to  the  conditions  and  range  for  which  they  have  been  found 
true,  and  for  which  n  is  apparently  constant.  This  is  more 
evident  when  the  wide  variation  of  the  value  of  such  constants 
as  determined  by  practice  is  considered.  Thus  Mr.  H.  G.  Reist 
gives,  as  the  practice  of  the  General  Electric  Company  on  generator 
bearings,  a  limiting  value  of  C  =  50,000  for  bearing  pressures  from 
30  to  80  pounds  per  square  inch.  Mr.  H.  P.  Been  gives  the 
practice  of  one  of  the  largest  Corliss  engine  builders  as  C  = 
60,000  to  78,000  for  bearing  pressures  not  higher  than  140 
pounds  per  square  inch. 

Unwin,  page  249,  gives  values  of  a  similar  constant,  ,#,  which 
corresponds  to  the  following  values  of  C: 

Locomotive  Crank  Pins 250,000  to  375,000 

Locomotive  Axles 200,000 

Marine  Engine  Crank  Pins 50,000  to    75,000 

Stationary  Engine  Crank  Pins i5>ooo  to    50,000 

Railway  Carriage  Axles 75,ooo  to  100,000 

Crank  Shaft  Bearings 7>5°o  to    20,000 

The  great  variation  in  these  values  of  C  is  no  more  than  might 
be  expected  in  view  of  the  foregoing,  and  also  in  view  of  the 
difference  in  lubrication  and  in  radiating  capacities  of  bearings, 
due  to  material,  form,  and  location.  While,  therefore,  these 
coefficients  may  form  a  guide,  and  while  doubtful  cases  may  be 
checked  for  heating  by  equation  (i),  care  should  be  exercised  that 
the  bearing  pressure  is  kept  within  the  limits  which  will  admit 
of  good  lubrication.  The  allowable  bearing  pressures  as  fixed 
by  practice  for  various  classes  of  machines  are  given  in  the  follow- 
ing table,  and  it  may  be  noted  that  these  are  more  accurately 


CONSTRAINING    SURFACES 


251 


known  than  the  values  of  />-,  or  the  values   of   the  coefficient  of 
radiation  K. 

Economy  in  the  use  of  material  and  the  importance  of  mini- 
mizing the  work  of  friction  suggest  that  the  diameter  of  the 
journal  shall  be  as  small  as  is  consistent  with  strength  and  stiff- 
ness. With  the  diameter  of  the  journal  determined  by  these 

TABLE  XIII 

BEARING    PRESSURES    FOR   VARIOUS    CLASSES    OF    BEARINGS 


CLASS  OF  BEARING  AND  CONDITION  OF  OPERATION. 


Allowable  Bearing  Pressure    in 
Ibs.  per  Square  Inch. 


Bearings  for  very  slow  speed  as  in  turntables  in 
bridge  work 

Bearings  for  slow  speed  and  intermittent  load  as  in 
punch  presses 

Locomotive  Wrist  Pins 

Locomotive  Crank  Pins 

Locomotive  Driving  Journals 

Railway  Car  Axles 

Marine  Engine  Main  Bearings  {  ^LmTractice 
Marine  Engine  Crank  Pins 

f     (high  speed) 
Stationary  Engine  Main  Bearings  j  for  dead  load.* 

[  for  steam  load . 
[       (high  speed) 
Stationary  Engine  Crank  Pins  j  overhung  crank .  . 

[  centre  crank 

Stationary  Engine  Wrist  Pins  (high  speed) 

[     (slow  speed) 
Stationary  Engine  Main  Bearings  \  for  dead  load.* 

[  for  steam  load . 

Stationary  Engine  Crank  Pins  (slow  speed) 

Stationary  Engine  Wrist  Pins  (slow  speed) 

Gas  Engines,  Main  Bearings 

Gas  Engines,  Crank  Pins 

Gas  Engines,  Wrist  Pins 

Heavy  Line  Shaft  Brass  or  Babbitt  Lining 

Light  Line  Shaft  Cast  Iron  Bearing  Surfaces 

Generator  and  Dynamo  Bearings 


7000  to  9000 


3000  to 

3000  to 

1500  to 

190  to 

300  to 

275  to 

400  to 

400  to 


4000 
4000 
1700 

22O 

325 
400 
500 
500 


60  to     1 20 
150  to    250 

900  to  1500 

400  to    600 

1000  to  1800 


80  to 

200  tO 

800  to 

1000  to 

500  to 

1500  to 

1500  to 

100  to 

15  to 

3°  to 


140 

400 
1300 

1500 

700 
1800 

2OOO 

150 

o5 

80 


considerations,  it  is  evident  that  the  length  of  the  journal  must 
be  such  that  the  bearing  pressure  is  within  the  allowable  limit. 
It  may  be,  however,  that  the  length  of  the  journal  thus  deter- 
mined wiD  be  so  great  that  localized  pressure  may  result;  or  it 
may  be  that  the  type  of  machine  will  not  allow  space  enough  for 


Weight  of  shaft,  flywheels,  etc. 


252  MACHINE    DESIGN 

such  a  length  of  bearing.     In  such  cases  the  diameter  must  be 

made  larger  and  the  length  may  be  correspondingly  decreased. 

While  practice  shows  wide  variations,  it  is  found  that  the  ratio 

of  the  length  of  the  journal  to  its  diameter  f  — )   is  fairly  well 

defined  for  any  given  class  of  machinery.  It  often  occurs,  there- 
fore, that,  when  journals  are  designed  with  the  ratio  as  fixed  by 
practice,  they  have  an  excess  of  strength  while  barely  satisfying 

the  conditions  as  to  bearing  pressure. 

/ 
The  following  are  average  values  of  —  as   found   in  good 

practice : 

TABLE  XIV 


TYPE   OF    BEARING 

Values  of  —r 

Marine  Engine  Main  Bearings  

I 
I 

I 

2 

3 

to  i  .5 
to  1.5 
to  2.5 
I 
to  1.5 
to  3 
to  4 
3 

Marine  Engine  Crank  Pins  

Stationary  Engine  Main  Journals  

Stationary  Engine  Crank  Pins  

Stationary  Engine  Crosshead  Pins 

Ordinary  Heavy  Shafting  with  Fixed  Bearings  .... 
Ordinary  Shafting  with  Self-adjusting  Bearings  .  .  . 
Generator  Bearings  

99.  Summary.     From  the  foregoing  the  following  statements 
may  be  made  regarding  imperfectly  lubricated  journals : 

(a)  The  minimum  diameter  of  a  journal  is  fixed  by  the  con- 
siderations of  strength  and  stiffness  under  the  loads  applied. 

(b)  The  smaller  the  diameter  of  the  journal  for  a  given  co- 
efficient of  friction,  the  less  is  the  work  of  friction  and  consequent 
liability  to  heating. 

(c)  The  tendency  of  the  bearing  to  heat,  other  things  equal, 
is  not  materially  affected  by  changing  the  diameter  of  the  jour- 
nal, but  is  reduced  by  increasing  the  length. 

(d)  The  projected  area  of  the  journal  must  be  such  that  the 
bearing  pressure  will  be  kept  within  the  allowable  limits  for  the 
particular  conditions;   and  the  ratio  of  length  to  diameter  must 
not  be  so  great  that  severe  localization  of  bearing  pressure  is 
liable   to   result.     These   considerations   may   require   a   larger 
bearing  than  the  previous  requirements  alone  would  demand. 


CONSTRAINING    SURFACES 


253 


(e)  The  work  of  friction,  per  unit  area,  must  not  exceed  the 
rate  of  radiation,  per  unit  area,  for  the  allowable  bearing  tempera- 
ture. 

100.  Perfectly  Lubricated  Journals.  It  was  shown  in  Article 
33  that  if  a  journal  is  supplied  with  sufficient  lubricant,  of 
proper  viscosity,  the  journal  itself  may  draw  in  the  lubricant  till 
a  film  is  formed  under  such  a  pressure  that  the  load  will  be  en- 
tirely fluid-borne.  With  any  given  set  of  conditions,  therefore, 
and  perfect  lubrication,  a  definite  journal  velocity  will  permit 
the  carrying  of  a  definite  load  per  unit  area  upon  the  journal, 
and  once  the  relation  is  established  between  the  load,  velocity, 
and  coefficient  of  friction,  it  is  constant,  and  not  unstable,  as 
in  the  case  of  imperfectly  lubricated  surfaces. 

It  was  further  shown  that  the  following  statements  are  true 
regarding  perfectly  lubricated  surfaces. 

(a)  The  friction  of  perfectly  lubricated  surfaces  for  a  given 
velocity  depends  very  little  on  the  materials  which  form  the 
rubbing  surfaces,  but  does  depend  largely  on  the  character   of 
the  lubricant. 

(b)  The  frictional  resistance  of  perfectly  lubricated  journals 
for  any  given  velocity  is,  within  the  limits  of  pressure  under  which 
the  oil  film  may  be  maintained,  independent  of  the  pressure; 
(that  is,  p-  w  =  a  constant) . 

(c)  The  coefficient  of  friction  of  perfectly  lubricated  surfaces, 
for  any  given  pressure,  varies  very  nearly  as  the  square  root  of 
velocity,  for  velocities  up  to  500  ft.  per  minute;    approximately 
as  the  fifth  root  of  the  velocity  for  velocities  between  500  and 
2,000  ft.  per  minute;  and  is  practically  independent  of  the  velocity 
for  values  above  2,000  ft.  per  minute. 

The  first  of  these  statements  has  been  abundantly  verified 
by  experiment  and  is  discussed  more  fully  in  Article  90. 

The  following  table,  which  is  one  of  the  several  given  in 
Tower's  report,  shows  clearly  the  truth  of  the  second  statement, 
for  the  frictional  resistance  is  seen  to  remain  practically  constant 
with  all  loads  at  any  fixed  velocity.  The  frictional  resistance 
is  also  seen  to  vary  very  nearly  as  the  square  root  of  the  velocity. 
Table  VIII,  Article  33,  which  was  deduced  from  Table  XV,  shows 


254 


MACHINE    DESIGN 


the  coefficients  of  friction  for  the  range  of  pressures  and  velocities 
given,  the  latter  not  exceeding  500  ft.  per  minute. 


TABLE  XV 

BATH   LUBRICATION 


jjlg 

Frictional  resistance  in  pounds  per  square  inch  of  projected  area  of  bearing  sur- 
face =  n  w,  for  velocities  in  feet  per  minute  as  below.    Temperature  =  90°  F. 

JfcrS 

105  ft. 

J57  ft- 

209  ft. 

262  ft. 

314  ft. 

366  ft. 

419  ft. 

471  ft. 

573 

•583 

.62 

.678 

.721 

•758 

•794 

Seized 

520 

.496 

•546 

•597 

.648- 

.691 

•735 

.771 

4i5 

.386 

•445 

-495 

•539 

.582 

.619 

363 

.306 

•35 

.401 

•444 

.488 

•S32 

.561 

258 

.277 

•357 

.416 

•459 

•5°3 

•547 

•583 

.626 

153 

.248 

.306 

•364 

.408 

•459 

.510 

.561 

.605 

IOO 

.277 

•357 

•423 

•503 

•576 

.619 

.663 

.714 

Tower's  experiments  at  different  temperatures  show  that  the 
coefficient  of  friction,  for  the  above  range  of  pressure  and  veloci- 
ties, decreases  as  the  temperature  increases.  His  principal  experi- 
ments, from  which  Table  XV  is  taken,  were  conducted  at  90°  F. 
and  without  artificial  means  of  cooling  the  bearing.  The  differ- 
ence between  the  coefficients  of  friction  obtained  at  90°  F.  and 
those  obtained  at  temperatures  as  high  as  are  usually  allowed  in 
practice,  can  be  neglected,  as  far  as  designing  is  concerned, 
especially  since  those  at  90°  are  on  the  safe  side.  Tables  VIII  and 
XV  may,  therefore,  be  taken  as  representing  fairly  well  the  rela- 
tion existing  between  pressure,  velocity,  and  frictional  resistance  for 
this  range,  which  fortunately  covers  the  most  usual  conditions 
in  practice.  It  is  to  be  noted  that  at  the  greatest  pressure  and 
highest  velocity,  the  bearing  seized,  indicating  that  with  such 
velocity  a  lower  pressure  must  be  assigned,  if  a  perfect  oil  film 
is  to  be  maintained,  or  that  with  this  greatest  load  a  lower  velocity 
must  be  assigned,  if  the  bearing  is  to  radiate  the  heat  of  friction. 
The  work  of  friction  at  471  ft.  per  minute  with  a  load  of  520  Ibs. 
is  seen  to  be  .771X471=365  ft.  Ibs.  per  minute,  or  about  6  ft. 
Ibs.  per  second.  From  curve  I,  Fig.  84  (a),  it  appears  that  to 
radiate  this  amount  of  energy,  the  bearing  must  attain  a  tempera- 


CONSTRAINING   SURFACES  255 

ture  of  over  110°  F.  above  that  of  the  surrounding  atmosphere,  or 
a  total  temperature  of  at  least  i8o°F. 

It  is  to  be  especially  noted  that  within  the  limits  of  pressure 
where  a  perfect  oil  film  will  form,  the  frictional  resistance,  for  a 
given  velocity,  is  practically  constant  and  independent  of  the 
pressure.  (See  Table  XV.) 

The  frictional  resistance,  and  coefficient  of  friction,  for  bear- 
ings running  at  velocities  of  over  2,000  ft.  per  minute  with  perfect 
lubrication,  have  been  quite  fully  determined  by  Wm.  O.  Lasche.* 
The  experimental  work  was  very  extensive,  the  results  very  con- 
clusive, and  should  be  carefully  read  by  designers  of  high-speed 
machinery.  A  discussion  of  these  experiments  is  beyond  the 
scope  of  this  treatise,  but  a  few  of  the  most  important  results  will 
be  considered.  Lasche  found  that  at  these  high  velocities  the 
coefficient  of  friction  was  practically  independent  of  the  velocity, 
but  varied  inversely  with  the  pressure  as  in  the  Tower  experi- 
ments, and  also  varied  inversely  with  the  temperature.  He  found 
that  if  w  be  the  bearing  pressure  in  pounds  per  square  inch,  and  t 
the  temperature  of  the  bearing  in  Fahrenheit  degrees,  then 

nw(t  -  32)  =  51.2      .      .    '.      .      .      (2) 
51.2 

"•""••(T^  ||  •  -.;   •    •   <s) 

For  velocities  between  500  and  2,000  ft.  per  minute  the  co- 
efficient of  friction  varies  about  as  the  fifth  root  of  the  velocity,  as 
shown  by  the  experiments  of  Stribeck.  As  far  as  designing  is  con- 
cerned, the  difference  between  the  coefficients  for  this  range  and 
those  found  by  Lasche  for  the  higher  velocities,  may  be  neglected, 
and  Lasche's  equation  may  be  applied,  without  serious  error,  to 
all  velocities  above  500  ft.  per  minute. 

Equation  (3)  may  be  written 


where  V  is  the  rubbing  velocity  in  ft.  per  minute.  Since  /'  w  V 
is  the  frictional  loss  per  unit  of  projected  area  in  foot  pounds  per 
minute,  equation  (4)  may  be  used  to  compute  the  heat  which  a 

*  See  Traction  and  Transmission,  January,  1903. 


256  MACHINE    DESIGN 

perfectly-lubricated  high-speed  journal  can  radiate  per  square 
inch  of  projected  area,  and  not  rise  above  a  temperature  /. 

The  limiting  values  of  the  pressure  under  which  a  perfect  oil 
film  can  be  maintained,  at  these  high  velocities,  have  not  been 
fully  determined.  In  Lasche's  experiments  a  load  of  213  pounds 
per  square  inch  of  projected  area  was  carried  at  a  velocity  of 
1,968  ft.  per  minute.  In  Kingsbury's*  experiments  loads  from 
80  to  86  pounds  per  square  inch  were  repeatedly  carried  at  velo- 
cities up  to  1,990  ft.  per  minute.  In  both  Kingsbury's  and 
Lasche's  work  either  the  oil  circulation,  or  the  bearing  it- 
self, was  artificially  cooled,  thus  materially  assisting  the  radia- 
tion. 

The  values  given  by  these  experiments  were  obtained  on 
experimental  machines  and  may  be  looked  upon  as  limiting 
values.  Successful  practice  in  the  design  of  steam  turbine  bear- 
ings gives  velocities  ranging  from  1,800  to  3,000  ft.  per  minute, 
with  pressures  inversely  as  the  velocity  ranging  from  80  to  50 
pounds  per  square  inch.  Where  the  pressure  is  as  high  as  90 
pounds  per  square  inch,  it  is  found  that  the  velocity  must  be 
kept  below  1,800  ft.f  per  minute.  The  empirical  equation 
•w  V  =  150,000  is  much  used  for  this  class  of  work,  and  gives 
values  agreeing  with  those  just  quoted.  It  is  evident  that  with 
these  high  velocites  the  radiation  must  be  assisted.  Thus  let  V 
=  2,000  and  w  =  75  in  accordance  with  the  empirical  rule  just 
.given,  and  let  it  be  required  to  keep  the  temperature  t  at  150°  F. 
or  a  temperature  of  say  75°  F.  above  the  atmosphere. 

Then  by  equation  (4)  the  frictional  work  is, 

ti  w  V  =  -r1 '—^r  =  867  ft.  Ibs.  per  minute  or  14.5  ft.   Ibs. 

(150  —  32) 

per  second,  whereas  the  bearing  alone,  if  connected  to  a  heavy 
iron  frame,  will,  from  Curve  2,  Fig.  84  (a),  radiate  only  6.4  ft.  Ibs. 
per  second.  Since  the  specific  heat  of  both  water  and  oil  are 
known  the  supply  of  either  necessary  to  carry  off  the  excess 
heat  of  friction  can  be  calculated. 

*  Transactions  A.  S.  M.  E.,  Vol.  XXVII,  page  425. 
t  See  "  Steam  Turbines,"  by  Frank  Foster,  page  181. 


OF  THE 

UNIVERSITY   1 

^STRA 


SURFACES 


257 


It  is  to  be  noted  that  with  perfect  lubrication  the  product  A*  w 
for  any  velocity  is  a  constant  quantity.  It  follows  therefore  that 
for  any  given  total  load  W  the  unit  bearing  pressure  should  be 
kept  as  high  as  possible  provided  it  does  not  exceed  the  max- 
imum allowable  value  for  the  given  conditions.  For  if  the  unit 
bearing  pressure  is  decreased,  either  by  increasing  the  diam- 
eter or  length  of  the  bearing,  the  coefficient  of  friction  is  cor- 
respondingly increased;  hence  the  total  frictional  resistance  n  W 
is  also  increased.  Care  should,  of  course,  be  exercised  in  any 
case  that  the  heat  of  friction  is  properly  carried  away. 

101.  Examples  of  Journal  Design.  Journals  generally  form 
an  integral  part  of  a  shaft  or  spindle,  and  the  determination  of 
the  stresses  acting  upon  them  is  a  part  of  the  solution  of  the 


tl 


1*--=- 


FIG.  86. 


stresses  in  the  shaft  itself.  It  is  desirable,  however,  to  point  out 
some  of  the  special  features  of  journal  design. 

The  actual  distribution  of  pressure  over  a  journal,  in  the 
direction  of  the  axis,  is  not  known;  but  there  is  every  reason  to 
believe  that  the  distribution  is  fairly  uniform.  Thus  bearings, 
as  a  rule,  wear  quite  uniformly  over  their  entire  length,  where 
fair  alignment  is  maintained.  It  is  customary,  in  the  absence 
of  exact  data,  to  assume  for  computations  as  to  strength  and 
rigidity  that  the  load  on  the  journal  is  concentrated  at  the 
middle  of  its  length.  This  assumption  is  on  the  safe  side,  and 
will  sometimes  give  shaft  diameters  excessively  large  as  far  as 
strength  is  concerned. 

The  following  examples  (a,  b  and  c)  show  the  most  important 
cases  of  journal  design.  It  is  assumed  in  each  case  that  the  bear- 


258  MACHINE    DESIGN 

ings  are  imperfectly  lubricated,  which  is  the  most  common  con- 
dition, but  the  application  of  the  theory  to  perfectly  lubricated 
journals  is  obvious. 

Example  (a)  .  This  case  is  illustrated  in  Fig.  85  .  Here  the  cen- 
tre of  the  bearing  is  fixed  at  O,  by  the  construction  of  the  machine. 
The  centre  line  of  the  pulley  M  is  also  fixed  at  XX,  by  the  location 
which  the  belt  must  occupy,  so  that  the  pulley  overhangs  the 
bearing  by  the  distance  a.  The  diameter  of  the  pulley  d  is  40 
inches,  a  =  io  inches,  the  pull  on  the  tight  side  of  the  belt  is  500 
Ibs.,  and  the  pull  on  the  slack  side  is  300  Ibs.  It  is  required  to 
determine  the  dimensions  of  the  journal. 

The  stresses  induced  in  the  journal  are,  torsional  stress  due 

d 
to  the  twisting  moment  (7\  —  T2)  —  ,  flexural  stress  due  to  the  bend- 

ing moment  (7\  +  T2)  a,  and  shear  due  to  the  direct  pull  7\  +  T2. 
The  last  is  small  and  is  usually  neglected  (see  Art.  26),  and  the 
journal  may  be  considered  as  subjected  to  a  combined  bend- 
ing and  twisting  moment.  Formula  K2  or  K3  (page  49),  there- 
fore, applies. 

The  bending  moment  M  =(T1  +  r2)  a  =  (500  +  300)  10  =  8,000 

d 
The  twisting  moment  T  =  (7\  —  T2)  -  =  (500  —  300)  20  =  4,000 

Hence  --  =  -      -  =  2  =  x  and  taking  —  =  .8  it  is  found  from 
T       4,000  p 

Figure  9  that  equation  K2  applies 


.'.  Me  =  -[x  +  Vx>  +  i]T  =  -[2  +  V7TT]  4,ooo  =  8,480 

2  2 

pi       p  7i  d3 
From  equation  /,  page  94,  M  t  =  —  =  -  . 

ord-.H^.  3121^480  _  8.63 

7i  p         n  X  10,000 

.'.  d  =  2>^  inches  (nearly),  or  say  2%" 
If  the  length  of  the  bearing  be  taken  at  7  inches  (see  Table  XIV), 

the  bearing    pressure  will    be  -  -  =  50  Ibs., 

2/4  X  7          I5  *75 
which  is  a  safe  value. 


CONSTRAINING    SURFACES  259 

If  the  number  of  revolutions  be  300  per  minute,  and  />.  be 
taken  as  .015,  the  work  of  friction  per  unit  of  projected  area  will 

n  X  2.25 
be  50  X  .015  X  300  X =  133  ft.  Ibs.  per  mm.,  or  2.2 

ft.  Ibs.  per  sec.  From  Curve  i,  Fig.  84  (a),  it  is  seen  that  to  radiate 
this  amount  of  energy  the  temperature  of  the  bearing  will  rise 
about  50°  above  the  surrounding  air.  This  is  a  safe  value  and 
the  design  is  satisfactory. 

Example  (b) .  Let  the  line  of  action  of  the  load  pass  through 
the  centre  line  of  the  journal,  as  in  the  case  of  the  steam-engine 
crank  pin  in  Fig.  86.  Let  the  length  of  the  crank  be  18  inches, 
and  the  total  maximum  pressure  on  the  crank  pin  be  25,000 
pounds.  What  should  be  the  dimensions  of  the  crank  pin  in 
order  to  be  safe  against  rupture  and  overheating? 

Referring  to  Table  XIV,  it  is  seen  that  journals  of  this  charac- 
ter are  short  compared  to  their  diameter,  and  hence  are  usually 
strong  enough  and  stiff  enough  if  designed  for  a  bearing  pressure 

low  enough  to  prevent  overheating.      Let  -;  be  taken  as  1.25. 

d 

From  Table  XIII  it  is  seen  that  900  Ibs.  per  square  inch  may  be 
safely  carried  on  this  type  of  pin.  If  d  be  the  diameter  of  the 
pin  and  /  the  length,  then  the  projected  area  of  the  pin  is  lxd  = 
1.25  dXd=i.2$d2. 

Whence  i .  2 $d2  X  900  =  25 ,000 
or  d2  =  22.2 

.  '  .  ^  =  4.7  or  say  5  inches, 
and  £  =  5X1.25  =6.25  inches. 

The  pin  may  now  be  checked  for  strength.  In  a  short  pin  of 
this  kind  it  is  more  accurate  to  assume  the  load  uniformly  dis- 
tributed along  the  pin,  than  to  assume  it  as  concentrated  at  the 
middle.  The  pin  may,  therefore,  be  considered  as  a  cantilever 
uniformly  loaded  with  a  load  W  =  25,000. 

Whence  from  Table  I,  case  3,  the  maximum  bending  moment 

Wl        25,000  X  6.25 
M   =  -  -  =  78,125  inch  pounds 


260  MACHINE    DESIGN 

pi  Me       32  M 

. ' .  from  equation  7,  page  94,  M  =  —  or  p  =  — -  =    ^ 

or  />  =  —  —     3       =  6,400    pounds    nearly,  which    is   a   safe 

value. 

In  a  similar  way  the  pin  may  be  checked  for  deflection,  if 
desired,  by  means  of  case  3,  Table  I. 

Example  (c) .  Sometimes  the  location  of  the  bearing  is  depend- 
ent on  the  diameter  of  the  shaft,  which  is  unknown,  and  in  such 
case  a  tentative  method  must  be  adopted.  Thus  in  Fig.  86  neither 
the  length  of  the  bearing  B,  nor  the  thickness  of  the  crank  hub  t, 
can  be  definitely  decided  upon  till  something  is  known  about  the 
diameter  of  the  journal.  The  diameter  must  therefore  be  as- 
sumed, and  then  checked  by  the  equations  which  apply  to  the 
case.  Usually  a  close  estimate  can  be  made  from  existing  ma- 
chines of  similar  type.  In  the  case  of  the  steam-engine  shaft, 
for  example,  it  is  known  that  the  main  journal  is  frequently  about 
one-half  the  diameter  of  the  cylinder.  The  data  taken  in  example 
(b)  correspond  to  a  cylinder  diameter  of  about  18  inches,  and  the 
journal  diameter  may  therefore  be  assumed  as  9  inches.  From 
Table  XIV,  the  length  of  the  journal  may  be  taken  as  20  inches. 
The  length  of  the  hub  should  be  at  least  8  inches,  for  this  di- 
ameter. The  boss  under  the  pin  may  be  taken  as  y%"  in  height 
and  since  the  pin,  from  case  (b),  is  6.25  inches  long,  the  total 
distance  from  the  centre  of  the  crank  pin  to  the  centre  of  the  shaft 
may  be  assumed  as  21 K  inches.  The  projected  area  of  the 
journal  is  9X20  =  180  square  inches,  which  gives  a  bearing 

pressure  of  — ^—  =  140  pounds  per  square  inch;  and  from  Table 

XIII  it  is  seen  that  this  is  a  safe  value  as  far  as  the  load  due  to  steam 
pressure  is  concerned.  If  the  shaft  also  carries  a  heavy  fly-wheel 
this  must  be  taken  into  account  (see  next  chapter). 

The  stresses  induced  in  the  journal  are  of  the  same  character 
as  in  case  (a).  Taking  the  length  of  crank  /  =  18  inches,  and  the 
pressure  on  the  pin  =  2 5,000  as  before,  then  the  bending  moment 
M  =  25,000 X  21  K  =  537-500  inch  pounds,  the  twisting  moment 


CONSTRAINING    SURFACES  261 

T  =  25,000  X  1 8  =  450,000  inch  pounds,  whence    —  =  1.19  = 

x,  and  taking  -r  =  .8,  equation  K2  is  found  by  Fig.  9  to  apply  to 
the  case. 

• '  •  M*  =  \  [*  +  W  +  i]T  =  I  [i .  19  +  Vi .  i92  +  i]  450.000 
=  616,500. 

77  T/  •  IN,  32    ^e  32X6l6,500 

From/  (as  in  example  a)  p  =  -  — — -  =  -  — y —  =  8,600, 

7T  &  7T   X  9 

which  is  a  safe  value  and  the  design  is  satisfactory. 

1 02.  Lubrication  of  Journals.  The  point  of  application  of 
the  lubricant  is  of  utmost  importance,  and  the  method  of  supplying 
the  lubricant  to  the  journal  sometimes  materially  affects  the 
design  of  the  bearing.  The  most  common  methods  of  feeding 
lubricants  to  rubbing  surfaces  as  given  in  Article  30  apply  fully  to 
journals  and  may  be  classified  as  follows: 


Imperfect  Lubrication 


Common  oil  hole. 

Common  wick  or  siphon  feed  cup. 

Common  drop  sight  feed  cup. 

Oily  pad  against  journal. 

Ring  or  chain  feed. 

Centrifugal  oiler. 

Compression  grease  cup. 


I  Bath  lubrication. 

Perfect  Lubrication  i  Flooded  lubrication. 
I  Forced  lubrication. 

In  flooded  lubrication  (sometimes  erroneously  called  forced 
lubrication) ,.  the  oil  is  supplied  to  the  bearing  under  a  low  pressure 
which  insures  that  the  journal  is  always  flooded  at  the  point  of 
application,  as  in  bath  lubrication,  but  it  does  not  force  the 
lubricant  between  the  surfaces.  In  forced  lubrication  the  oil  is 
supplied  at  a  pressure  in  excess  of  the  film  pressure  at  the  point 
of  application,  and  is  thus  forced  in  between  the  surfaces,  no 
reliance  being  placed  on  the  tendency  of  the  journal  to  draw  in 
the  lubricant.  The  compression  grease  cup,  while  supplying 
the  lubricant  under  slight  pressure,  gives  only  imperfect  lubrica- 


262  MACHINE    DESIGN 

tion  as  the  supply  of  lubricant  is  not  copious  as  in  the  case  of 
forced  lubrication. 

In  applying  any  of  these  methods  of  lubrication,  therefore, 
except  the  compression  grease  cup  and  forced  lubrication,  care 
should  be  exercised  that  the  point  of  application  is  at,  or  near,  a 
point  of  lowest  pressure  and  at  the  place  where  the  journal  will 
naturally  draw  in  the  lubricant.  Thus,  in  Fig.  80,  if  the  pres- 
sure is  always  downward,  lubricant  can  be  supplied  at  H  for 
motion  in  either  direction.  If  the  pressure  were  upward,  an 
oil  hole  at  H  would  not  only^be  useless  for  supplying  lubricant 
but  would  be  fatal  to  good  lubrication,  as  any  tendency  for  a 
pressure  film  to  form  would  be  destroyed  by  relief  of  the  pressure 
at  the  hole.  In  such  a  case  the  lubricant  should  be  supplied 
from  underneath,  or  if  the  direction  of  rotation  were  anti-clock- 
wise an  oil  hole  as  shown  at  /  would  be  good  design.  In  forced 
lubrication  the  point  of  application  should  be  the  point  of  greatest 
bearing  pressure,  and  the  hydraulic  pressure  under  which  the  oil 
is  supplied  should  be  greater  than  the  maximum  bearing  pressure. 

While  the  decreased  friction  due  to  perfect  lubrication  is 
evident,  it  does  not  follow  that  an  effort  should  be  made  to  design 
every  bearing  so  as  to  secure  this  advantage.  In  some  places  a 
simple  oil  hole  is  sufficient,  in  others  a  constant  supply  from  a 
wick  feed  will  suffice,  while  again,  with  greater  speeds,  a  ring 
oiling  device  is  necessary.  In  many  modern  power  instal- 
lations, with  either  steam  turbines  or  reciprocating  engines, 
very  complete  apparatus  for  supplying  flooded  lubrication  will 
be  found.  The  bearings  are  constructed  so  as  to  catch  all  the 
oil,  as  it  leaves  the  journal,  and  pipes  convey  it  to  a  central 
receiver.  A  pump  continually  circulates  the  oil  to  the  various 
bearings,  and  in  the  best  installations  the  oil  is  filtered  and  cooled 
during  the  circuit.  The  same  results  are  obtained  by  flooded 
lubrication  as  with  bath  lubrication.  Forced  lubrication  is 
resorted  to  only  where  the  bearing  pressures  are  excessive  and 
beyond  those  which  can  be  supported  by  the  natural  action  of 
the  film  formed  by  rotation  of  the  journal.  (See  Art.  33.) 

The  location  and  character  of  the  oil  grooves  deserve  special 
attention.  If  the  velocity  of  the  journal  is  so  low  as  to  draw  in 


CONSTRAINING    SURFACES  263 

little  lubricant  the  oil  grooves  should  be  so  cut  as  to  allow  the 
lubricant  to  flow  in  near  the  points  of  greatest  pressure.  Grooves, 
or  scores  on  the  journal  itself,  have  been  found  helpful  in  drawing 
in  the  lubricant  under  such  circumstances;  especially  where  the 
lubricant  is  heavy.  But  where  the  velocity  is  above  25  feet  per 
minute  (see  Fig.  16),  and  for  ordinary  pressures,  care  should  be 
used  that  no  oil  grooves  are  cut  that  will  tend  to  prevent  the 
formation  of  the  pressure  film.  If  the  lubricant  is  delivered  at 
H,  Fig.  80,  and  the  pressure  is  downward,  oil  grooves  of  any 
kind  running  from  H  which  will  distribute  the  oil  over  the  surface 
of  the  journal,  are  allowable  so  long  as  they  terminate  at  a  little 
distance  from  the  edge  of  the  bearing.  If  the  oil  is  delivered  at 
/,  and  the  pressure  is  either  downward  or  upward,  the  grooves 
should  be  cut  at  right  angles  to  the  direction  of  motion,  so  as  to 
distribute  the  oil  along  the  entire  length  of  the  bearing.  If  cut 
diagonally  they  will  extend  under  the  journal  toward  the  point  of 
greatest  oil  pressure,  thus  relieving  any  tendency  to  the  formation 
of  a  pressure  film  and  the  lubrication  will  not  be  as  good  as  it 
would  be  if  no  grooves  were  present. 

The  sharp  edges  of  all  oil  grooves  should  be  carefully  removed 
to  facilitate  the  passage  of  the  oil  under  the  journal.  The  sharp 
edges  of  the  bearings  themselves  should  also  be  filed  or  scraped 
away  for  the  same  reason.  Where  one  bearing  surface  encircles 
nearly  one-half  of  the  shaft,  as  in  Fig.  80,  the  surfaces  should  be 
relieved  for  some  little  distance  from  the  parting  line  to  help 
the  wedging  action  of  the  oil  and  to  insure  the  journal  against 
side  pressure  due  to  springing  of  the  bearing  under  the  load. 
A  bearing  which  binds  sidewise  will  not  lubricate  properly. 

THRUST  BEARINGS 

103.  General  Considerations.  When  a  shaft  is  subjected  to 
a  heavy  end  thrust,  either  from  the  weight  of  the  parts  carried 
or  on  account  of  the  power  transmitted,  the  simple  collars  which 
are  used  to  prevent  end  thrust  in  ordinary  shafting  will  not 
suffice,  and  bearings  of  special  form,  known  as  thrust  bearings, 
must  be  provided.  If  the  bearing  is  designed  so  that  the  thrust 
is  taken  on  the  end  of  the  shaft  it  is  called  a  step-bearing  or 


264 


MACHINE    DESIGN 


footstep-bearing.     If  the  thrust  bearing  must  be  placed  at  some 
distance  from  the  end  of  the  shaft  it  is  called  a  collar  bearing. 

104.  Step-Bearings.  If  the  motion  of  rotation  is  very  slow, 
as  is  the  case  in  swinging  cranes  and  similar  work,  a  simple  cast- 
iron  step,  as  shown  in  Fig.  87,  will  meet  the  requirements,  even 
if  the  pressure  is  heavy.  If,  however,  the  velocity  is  high,  this 
simple  arrangement  will  not  give  good  results,  even  when  the 
pressure  per  unit  area  is  low.  It  may  be  assumed,  without  great 
error,  that  the  unit  pressure  between  the  faces  of  a  newly  fitted 
step-bearing  is  uniform  at  all  points.  The  velocity  of  rubbing, 
however,  is  a  maximum  at  the  outer  edge,  and,  theoretically,  it  is 
zero  at  the  geometric  centre  of  the  pivot.  Since  the  wear  is  pro- 
portional to  the  product  of  pressure  and  velocity,  it  follows  that 

the  surface  will  wear  unevenly, 
the  greater  wear  taking  place 
at  the  outer  edge.  This  will 
bring  a  concentrated  pressure 
at  other  points,  and  heating  and 
cutting  may  result.  It  is  always 
advisable  in  heavy  work,  for  this 
reason,  to  remove  the  wearing 
surface  near  the  center,  where 
the  motion  is  slowest,  and 
where  eventually  the  greatest  concentration  of  pressure  is  likely 
to  be  produced  (see  Fig.  87).  Decreasing  the  bearing  pressure 
by  increasing  the  surface,  is  effective  within  limits,  since  the 
area  increases  as  the  square  of  the  diameter  while  the  velocity 
of  rubbing  increases  directly  as  the  diameter.  Increasing  the 
radius,  however,  increases  the  average  moment  arm  of  the 
frictional  resistance,  and  hence  increases  the  lost  energy.  It  is 
often  better,  therefore,  to  carry  a  higher  bearing  pressure,  and 
thus  keep  the  diameter  of  the  pivot  small. 

If  a  number  of  discs  are  placed  between  the  step,  or  pivot, 
and  the  bearing  (Fig.  88),  they  have  the  effect  of  reducing  the 
relative  velocity  between  adjacent  surfaces;  and  if  the  rotative 
velocity  of  the  pivot  is  high,  they  are  very  useful  as  a  safeguard 
against  cutting;  for  if  abrasion  should  begin  between  any  pair 


FIG.  87. 


FIG. 


CONSTRAINING    SURFACES 


265 


of  discs,  motion  will  cease  at  that  point  till  the  lubrication 
became  effective  again.  These  washers  are  usually  made  alter- 
nately of  steel  and  brass,  or  some  other  metal,  and  the  upper  and 
lower  washers  are  fastened  to  the  shaft  and  bearing  respectively. 
An  oil  hole  passes  through  the  centre  of  the  washers,  and  radial 
grooves  cut  across  the  faces  permit  a  flow  of  oil  between  the 
surfaces,  centrifugal  action  assisting  the  lubrication.  If  the  top 
of  the  bearing  is  connected  to  the  bottom  by  an  oil  passage,  as 
shown  at  N  (Fig.  88),  the  centrifugal  action  will  set  up  a  con- 
tinuous circulation  of  the  oil,  making  the  lubrication  effective. 
The  unit  pressure  between  washers  is  the  same  as  between  the 
shaft  and  the  first  washer,  but  the  relative  motion  between  the 
surfaces  is  decreased  and  the  wear  thus  reduced.  A  combination 


FIG.  89. 


FIG.  90. 


FIG.  91. 


of  hardened  and  ground  steel  washers,  alternating  with  brass  or 
bronze  washers,  makes  an  effective  bearing.  Sometimes  the 
washers  are  made  lenticular  in  shape,  as  shown  in  Fig.  89,  in 
order  to  allow  the  shaft  automatically  to  adjust  its  alignment. 
For  very  light  work  the  shaft  sometimes  rests  on  a  pair  of  hardened 
steel  buttons,  or  a  hardened  steel  ball  which  runs  between  hardened 
steel  surfaces  is  introduced.  In  the  submerged  step-bearings  of 
water  turbines  the  shaft,  which  is  often  capped  with  bronze,  rests 
on  a  lignum  vitae  step  and  lubrication  is  effected  by  the  surround- 
ing water. 

If  the  outline  of  a  step-bearing  be  made  that  of  a  tractrix* 
(Fig.  90) ,  it  is  found  that  the  tendency  to  wear  in  an  axial  direction 


*  See  Church's  "  Mechanics,"  page  181. 


266 


MACHINE    DESIGN 


is  uniform  at  all  points;  in  fact  if  two  homogeneous  flat  surfaces 
are  rotated  together  they  tend  to  wear  into  the  form  of  a  tractrix 
as  has  been  proven  by  experiment.  This  is,  therefore,  the  correct 
shape,  theoretically,  for  all  step-bearings;  but  on  account  of  the 
difficulty  and  expense  of  machining  the  surfaces,  it  is  seldom 
used.  The  tractrix  has  been  called  Schiele's  Anti-friction  Curve 
after  the  discoverer  of  the  above  property.  This  is  a  misnomer, 
however,  for  the  friction  of  a  tractrix-shaped  step  is  much  higher 
than  that  of  a  plain  pivot. 

It  is  evident  that  the  rubbing  surfaces  of  all  the  step-bearings 
which  have  been  discussed  can  be 
submerged  in  an  oil  bath.  The  lu- 
brication thus  obtained  is  not  to  be 
confused  with  that  obtained  on  hori- 
zontal rotating  bearings  discussed 
formerly.  While  centrifugal  force 
does  drive  the  oil  from  the  centre  to 
the  outside,  there  is  little  action  on  the 
part  of  the  surfaces  themselves  tend- 
ing, on  account  of  its  viscosity,  to 
draw  the  lubricant  between  them,  as 
in  horizontal  bearings.  Such  lubri- 
cation cannot  therefore  be  looked  on 
as  perfect  lubrication  although  giv- 
ing excellent  results.  The  experi- 
ments of  Beaucamp  Tower*  on  a 
steel  foot  step,  three  inches  in  diam- 
eter, gives  considerable  information  on  this  subject.  It  was 
found  that  a  single  diametral  oil  groove  was  better  than  more, 
and  pressures  up  to  160  pounds  per  square  inch  were  success- 
fully carried  at  128  revolutions  per  minute.  The  foot  step 
was  freely  lubricated,  and  rested  directly  on  the  bearing,  no 
washers  being  interposed.  At  240  pounds  per  square  inch  the 
bearing  seized. 

If  under  heavy  loads  the  maintenance  of  lubrication  is  im- 


THRUST  BEARING  OF  CURTISS 
VERTICAL  STEAM  TURBINE 

FIG.  92. 


*  Transactions  of  the  Institute  of  Mechanical  Engineers.  1891,  page  in. 


CONSTRAINING    SURFACES 


267 


portant,  the  lubricant  should  be  supplied  at  the  centre  of  the  step- 
bearing  under  a  pressure  such  that  the  metallic  surfaces  are 
forced  apart  and  the  load  is  fluid-borne.  Fig.  92  shows  a  recent 
form  of  the  step-bearing  used  on  the  Curtiss  steam  turbine.  The 
vertical  shaft  A ,  which  supports  the  heavy  rotating  parts  of  both 
turbine  and  generator,  is  carried  on  the  disc  B  which  rotates  with 
it.  The  lower  disc  C  can  be  adjusted  vertically,  by  means  of 
the  screw  E,  and  is  prevented  from  rocking  on  E  by  the  screws  F. 
Oil  is  forced  between  the  discs  through  the  central  pipe  Enforcing 
the  discs  apart  and  escaping  into  the  cavity  G.  The  load  is  thus 
completely  fluid-borne  and  perfect  lubrication  is  maintained. 


>0il  Holes 


FIG.  93. 

The  oil  passes  from  G  upward  through  the  guide  bearing  escaping 
atH. 

105.  Collar  Thrust  Bearings.  When  the  thrust  bearing  must 
be  placed  at  some  distance  from  the  end  of  the  shaft,  the  shaft 
is  provided  with  collars  integral  with  itself,  which  bear  against 
the  resisting  surfaces  as  shown  in  Fig.  93,  which  illustrates  a 
thrust  bearing  as  used  for  marine  work.  In  cheap  work,  or 
where  the  load  is  small,  a  single  collar  is  sometimes  used.  Occa- 
sionally a  series  of  washers,  as  in  Fig.  88,  are  interposed  between 
the  collar  and  the  bearing  ring.  The  objection  to  the  single- 
collar  bearing  for  heavy  loads  is  that  the  large  diameter  necessary 
to  obtain  a  practical  bearing  pressure  increases  the  work  of 
friction,  due  to  the  increased  velocity,  and  the  difference  between 
the  rubbing  velocities  of  the  ring  at  the  shaft  and  at  its  outer 
diameter  results  in  unequal  wear.  The  outer  diameter  of  the 
ring,  or  collar,  is  usually,  therefore,  not  more  than  one  and  one- 


268  MACHINE    DESIGN 

half  times  the  diameter  of  the  shaft,  which  limits  the  width  of 
face  of  the  collar  even  in  large  shafts  to  a  few  inches;  and  the 
necessary  area  is  obtained  by  using  a  number  of  rings. 

In  small  or  cheap  work,  the  bearing  surfaces  of  the  thrust  block 
are  sometimes  made  integral  with  the  bearing  proper;  but  usually 
they  are  made  detachable.  Thus  the  main  casting  of  the  block 
may  be  of  cast  iron  and  the  bearing  rings  of  brass  are  inserted  and 
held  in  place  by  radial  grooves  cut  in  the  block.  These  rings  must 
be  scraped  until  each  collar  on  the  shaft  bears  properly  against 
its  mating  ring,  so  that  the  thrust  is  uniformly  distributed.  The 
most  modern  practice  in  marine  work  is  to  make  the  bearing  rings 
horseshoe-shaped,  as  in  Fig.  91,  so  that  each  ring  can  be  with- 
drawn without  disturbing  any  other  portion  of  the  bearing  or 
shaft.  Occasionally  the  horseshoe  collars  are  adjustable  along 
the  shaft  so  as  to  be  more  easily  brought  to  a  proper  bearing.  In 
first-class  work  each  horseshoe  has  its  own  independent  water 
circulation,  so  that  local  heating  may  be  prevented,  and  the  lower 
part  of  the  bearing  constitutes  an  oil  bath  into  which  the  collars 
dip.  This  oil  bath  also  has  a  water  circulation  for  cooling  the  oil. 
106.  Friction  and  Efficiency  of  Thrust  Bearings.  If  P  be  the 
total  load  on  a  flat  circular  pivot  of  radius  r±  and  /*  be  the  co- 
efficient of  friction,  then  the  frictional  moment  resisting  rotation  is 

M  =  -p.Pr,*    ......     (i) 

\j 

.     If  r±  be  in  inches  and  P  be  in  pounds  then  the  energy  lost  per 
minute  in  foot  pounds  is 

£  =  -^Pr1x-       -=.  ^gp-Pr,N       .      .      (2) 
o 

where  N  is  the  number  of  revolutions  per  minute. 

In  a  similar  manner  if  the  thrust  be  taken  on  a  collar  of  out- 
side radius  rly  and  inside  radius  r2,  then 


(4) 


*  Church's  "  Mechanics,"  page  180. 


CONSTRAINING    SURFACES 


269 


The  efficiency  of  a  thrust  bearing  cannot  always  be  expressed 
as  a  function  of  the  power  transmitted.  Thus  in  the  case  of  a 
vertical  shaft  carrying  a  heavy  load  of  gears,  the  frictional 
resistance  of  the  step  has  little  to  do  with  the  power  transmitted. 
In  the  case  of  the  thrust  bearing  of  a  steamship  the  frictional 
moment  and  energy  loss  are  directly  proportional  to  the  driving 
force  P.  In  either  case,  however,  the  frictional  moment  or  the 
energy  loss  must  be  added  to  the  turning  moment  or  the  energy 
supplied,  as  the  case  may  be. 

The  following  coefficients  of  friction  are  taken  from  Tower's 
experiments : 

TABLE  XVI 


Coefficients  of  Friction    of   Flat   Pivots    for  the   Revolutions    per 

Pressures  in  Ibs.  per 

Minute  as  given  below. 

5oR.P.M. 

I28R.P.M. 

i94R.P.M. 

29oR.P.M. 

353  R.P-M. 

20 

.0196 

-      .0080 

.0102 

.0178 

.0167 

40 

.0147 

.0054 

.0061 

.0107 

.0096 

80 

.0181 

.0063 

.0045 

.0064 

.0063 

120 

.O22I 

.0083 

.0052 

.0048 

•°°53 

140 

.0093 

.0062 

.0046 

.0054 

At  50  and  128  R.P.M.,  the  oil  supply  was  restricted,  but  at 
the  other  velocites  the  bearing  was  flooded.  In  all  cases  the 
coefficient  increased  at  revolutions  below  40  R.P.M.,  which 
was  probably  due  to  the  decrease  of  the  centrifugal  force 
(the  bearing  being  oiled  from  the  centre).  This  would  seem 
to  indicate  that  devices  for  reducing  relative  rubbing  veloc- 
ity, such  as  multiple  washers  (Art.  102),  may  be  carried  to  an 
extreme,  causing  more  friction  than  a  plain  flat  pivot  where 
centrifugal  action  is  effective.  In  the  case  of  thrust  collars,  such 
as  shown  in  Fig.  91,  running  in  an  oil  bath,  the  surfaces  them- 
selves tend  to  draw  in  lubricant  in  a  way  similar  to  that  of  the 
ordinary  journal.  The  coefficients  of  friction  for  this  class  of 
thrust  should  therefore  be  as  low  at  least  as  those  given  above. 

107.  Bearing  Pressures  on  Thrust  Bearings.  Where  the 
velocity  of  rubbing  is  very  low  and  wear  is  not  important,  as  in 
the  case  of  swinging  cranes,  very  heavy  unit  loads  may  be  put 


270 


MACHINE    DESIGN 


upon  pivot  bearings,  especially  if  they  rotate  in  an  oil  bath. 
Where  the  velocity  is  high,  or  even  moderate,  and  wear  is  im- 
portant, much  lower  pressures  must  be  carried  with  imperfect 
lubrication,  than  on  ordinary  bearings  running  at  the  same 
velocity.  With  forced  lubrication,  as  in  the  step-bearing  shown 
in  Fig.  92,  it  is  evident  that  very  heavy  pressure  may  be  main- 
tained. If,  on  the  other  hand,  too  many  collars  are  used  on  a 
collar  thrust  bearing,  in  an  effort  to  keep  the  bearing  pressure 
down  to  a  low  value,  there  is  danger  that  all  of  the  collars  will 
not  bear  simultaneously.  The  following  are  average  values  of 
bearing  pressures,  for  thrust  bearings,  as  found  in  practice : 

TABLE  XVII 


Mean  Velocity  in  ft.  per  Min. 

Character  of  Lubrication. 

Bearing  Pressure  in  Ibs.  per 
Square  Inch. 

Very  slow  as  in  hand  cranes 
Up  to    soft. 
50  to  125 

125  tO  2OO 
200  tO  500 

500  to  800 

Bath  as  in  Fig.  87 
Bath  as  in  Fig.  88 
Bath  as  in  Fig.  88 
Bath  as  in  Fig.  88 
Bath  as  in  Fig.  88 
Thrust  Bearing  and  Bath 
Lubrication  as  in  Fig.  93. 

2,000  to  3,000 
200 
150 

100 

50 

75  to  5° 

Example.  Design  the  thrust  journal  for  a  steamship  having 
the  following  data,  and  estimate  the  frictional  loss  in  the  thrust 
bearing. 

Speed  in  knots  (i  knot=  6,080  ft.  per  hour) 15 

Indicated  horse-power  of  one  engine 5,ooo 

Inside  diameter  of  -thrust  collars 14" 

Outside  diameter  of  thrust  collars 21" 

Allowable  pressure  per  sq.  in.  of  surface 40  Ibs. 

Revolutions  of  the  shaft  per  minute 120 

Owing  to  frictional  losses  in  the  engine,  propeller,  and  shaft 
only  about  two-thirds  of  the  indicated  power  is  delivered  to  the 
thrust  block.  The  pressure  against  the  thrust  block  multiplied 
by  the  distance  through  which  the  ship  moves  per  minute  must 
equal  the  energy  delivered  to  the  block  per  minute;  or  if  P  be  the 
thrust,  S  the  speed  of  the  ship  in  knots  per  hour,  and  the  indicated 


horse-power  be  denoted  by  /.  H.  P.,  then  -X  I.  H.P.  X  33,000 


CONSTRAINING    SURFACES  271 

_PX5X  6,080  prj3     2  X  I.H.P.  X  33,000  X  60     7J?.P.  x  217 
60  3  X  5  x  6,080  5 

Hence    in  the  above  example  P  =  —      '-  =  72,300.    The 

0 

area  of  each  thrust  collar  =  -  (2i2—  i42)  =  192  sq.  in.      There 

4 

fore  the  total  allowable  pressure  on  each  collar   =192  X  40  = 
7,680  and  the  number  of  collars  =  =  9.5  or  say  10. 

If  the  bearing  runs  in  an  oil  bath,  the  coefficient  of  friction 
will  not  be  more  than  .01  under  the  worst  ordinary  conditions. 


Therefore  from  (4)     E=  .349 vPN 

i        i 

CIO    q;3_ ? 
10    52  - 

=  405,000  ft.  Ibs.  per  min.  or  12.3  H.P. 

ROLLER  AND  BALL  BEARINGS 

108.  General  Consideration  of  Rolling.  It  was  pointed  out 
in  Article  29  that  the  resistance  due  to  rolling  friction  was  much 
less  than  that  due  to  sliding  friction,  for  a  given  load.  The 
application  of  this  principle  to  very  heavy  loads  and  low  speeds, 
as  in  the  case  of  moving  heavy  bodies  en  rollers,  is  of  great 
antiquity;  but  only  in  recent  years  have  mechanics  been  able  to 
produce  surfaces  of  such  a  character  as  could  carry  even  very 
light  loads  at  high  speeds  on  either  roller  or  ball  bearings.  At 
present,  however,  bearings  of  this  character  can  be  obtained  which 
will  run  well  under  very  severe  conditions. 

When  a  curved  surface  rolls  upon  any  other  surface  with 
which  it  theoretically  makes  line  or  point  contact,  the  two  sur- 
faces tend  mutually  to  deform  each  other,  the  amount  of  deforma- 
tion depending  on  the  character  and  hardness  of  the  materials 
forming  the  surfaces,  and  the  intensity  of  the  load  sustained.  If 
the  surfaces  of  both  members  are  very  hard,  and  the  load  is  very 
light,  the  deformation  is  negligible  and  true  rolling  can  be  practi- 


272  MACHINE    DESIGN 

cally  attained.  When,  however,  any  appreciable  load  is  to  be 
carried  the  mutual  deformation  of  the  surfaces  destroys  the 
theoretical  line  or  point  contact  and  the  load  is  borne  on  a  small 
surface.  This  occurs  even  when  the  surfaces  are  very  hard,  and 
the  action  instead  of  being  that  of  pure  rolling,  is  a  combination 
of  rolling  and  sliding.*  The  true  theory  of  this  action,  which  is 
very  complex,  has  not  been  fully  demonstrated  and  is  beyond  the 
scope  of  this  treatise.  It  can  readily  be  seen  that  it  is  closely 
connected  with  the  elastic  properties  of  materials,  on  which  much 
research  work  has  been  done.  Undoubtedly  the  work  of  this 
character,  which  is  of  most  value  in  the  design  of  roller  or  ball 
bearings,  is  that  of  Professor  Stribeck  whose  masterly  report  has 
been  translated  into  English  by  Mr.  Henry  Hess,|  and  to  this 
translation  reference  will  be  made  hereafter. 

If  the  intensity  of  pressure  be  such  that  the  elastic  limit  of 
the  materials  is  exceeded,  permanent  deformation  will  occur. 
In  the  case  of  roller  or  ball  bearings  this  may  result  in  the  destruc- 
tion of  the  surfaces  either  by  flaking  off  locally,  or  by  simply 
crushing  out  of  shape.  In  either  case  continued  action  of  this 
character  is  destructive  to  the  bearing.  Experiments  on  either 
balls  or  rollers  to  determine  the  ultimate  crushing  load  are,  there- 
fore, misleading  and  useless  as  far  as  the  design  of  such  bearings 
is  concerned.  It  appears  from  experiment  and  experience  that 
bearings  of  this  character  can  be  constructed  to  carry  fairly 
heavy  loads  at  high  speeds  for  a  long  period  of  time  provided 
the  intensity  of  pressure  is  not  too  great.  It  is  obvious  from 
the  foregoing  that  the  materials  used  in  such  bearings  must  be 
homogeneous,  and  of  uniform  hardness.  The  success  of  the 
modern  ball  and  roller  bearing  has  been  made  possible  by  im- 
proved materials  and  workmanship  rather  than  by  new  theories. 

Referring  to  Fig.  94,  it  is  evident  that  when  two  adjacent 
rollers  or  balls,  A  and  B,  touch  each  other,  the  directions  of 
motion  of  the  common  points  of  contact  are  in  opposite  directions. 
It  is  often  stated  that  this  results  in  considerable  frictional  loss; 

*  The  student  may  demonstrate  this  action  by  rolling  a  round  lead  pencil  on 
a  piece  of  soft  rubber  under  pressure. 

f  See  Transactions  A.  S.  M.  E.,  Vol.  XXVIII. 


CONSTRAINING   SURFACES  273 

and  sometimes  small  intermediate  balls,  or  rollers,  are  used  as 
shown  at  C,  Fig.  94,  to  obviate  the  supposed  loss.  Such  inter- 
mediate balls  or  rollers  must  be  kept  in  place  by  a  cage  such  as 
E,  Fig.  94,  and  this  cage  will  give  rise  to  a  greater  frictional  loss 
than  that  which  it  is  expected  to  remedy.  A  brief  reflection  will 
show  that  very  little  pressure  can  possibly  exist  between  A  and  B. 
The  only  pressures  that  can  be  exerted  by  the  guiding  surfaces 
upon  the  balls  or  rollers  are  in  a  radial  direction  or  normal  to  the 
surfaces,  and  these  have  no  component  tending  to  force  the  ad- 
jacent rollers  or  balls  together.  Sometimes  the  rollers  or  balls  are 
separated  by  a  guiding  cage  (see  Fig.  95) ,  and  if  any  appreciable 
pressure  could  exist  between  adjacent  rollers  or  balls  the  same 
would  necessarily  exist  between  them  and  this  guiding  cage. 
This  theory  is  not  borne  out  by  experience,  as  these  cages,  in 
well-built  roller  bearings,  do  not  wear  appreciably.  The  fric- 
tional loss  from  this  source  is  undoubtedly  very  small. 

The  friction  of  roller  and  ball  bearings  while  at  rest  is  very 
small,  and  this  is  a  very  important  point  in  the  design  of  heavy, 
slow-moving  machinery  where,  with  ordinary  sliding  bearings, 
it  often  takes  a  much  greater  effort  to  start  the  machinery  from 
rest  than  to  maintain  motion  at  full  speed. 

ROLLER  BEARINGS 

109.  Forms  of  Bearings.  Roller  bearings,  in  common  with 
the  -ordinary  bearing,  are  classified  as  radial  or  thrust  bearings, 
according  to  the  manner  in  which  the  load  is  sustained.  A 
typical  form  of  construction  of  roller  bearings  for  radial  loading 
is  shown  in  Fig.  95.  A  shell  of  hardened  steel,  B,  surrounds 
the  shaft  A,  and  is  secured  firmly  to  it.  The  rollers  C  bear  against 
this  shell  B,  and  against  an  outer  shell  D,  which  is  secured  to  the 
bearing  proper,  E.  Both  rollers  and  shells  are  usually  made  of 
high  carbon  steel  hardened  and  ground,  or  of  mild  steel  case- 
hardened.  The  rollers  are  held  parallel  with  the  axis  of  the  shaft 
by  means  of  a  cage  F  which  is  made  of  brass  or  other  soft  material. 
Some  form  of  cage  is  necessary  in  all  roller  bearings  on  account 
of  the  tendency  of  the  rollers  to  twist  out  of  line  with  the  shaft, 
thus  replacing  the  theoretical  line  contact  with  point  contact  and 

18 


274 


MACHINE    DESIGN 


also  causing  an  end  pressure  and  cramping  on  the  rollers.  This 
tendency  to  end  thrust  is  sometimes  provided  for  by  putting  a 
small  ball  at  each  end  of  the  roller  to  act  as  a  thrust  bearing.  If 
the  axis  of  the  roller  is  not  parallel  to  that  of  the  shaft,  it  cannot 
make  line  contact  with  the  shaft  unless  it  assumes  a  spiral  form. 


FIG.  95. 


FIG.  94. 


If  the  surfaces  which  confine  the  roller  are  accurately  made,  and 
the  clearance  is  very  small,  as  it  should  be,  the  roller  cannot  get 
out  of  parallelism  with  the  shaft  without  being  bent  into  a  spiral 
form.  If  the  rollers  are  hardened  this  may  result  in  fracturing 
them,  especially  if  they  are  relatively  long.  To  obviate  this 


FIG.  960 


FIG.  97. 


trouble  the  rollers  are  sometimes  made  in  short  lengths,  as  shown 
at  H,  in  Fig.  95,  or  the  roller  is  made  flexible  as  illustrated  by  the 
Hyatt  roller  shown  in  Fig.  96.  This  roller  is  made  by  winding 
steel  strip  spirally  upon  a  mandrel,  thus  making  a  hollow  flexible 
roller.  It  is  to  be  especially  noted  that  neither  of  these  methods 
will  compensate  for  inaccurate  workmanship.  For  continuous 


CONSTRAINING    SURFACES 


275 


line  contact  the  outer  and  inner  shells  must  be  machined  with 
great  accuracy,  placed  in  very  accurate  alignment,  and  the  rollers 
must  be  guided  so  as  to  remain  perfectly  parallel  to  the  shaft. 
These  conditions  are  difficult  to  obtain  initially,  and  almost 
impossible  to  maintain  with  great  accuracy  under  continuous 
service.  The  rollers  in  bearings  for  radial  loading  may  be 
cylindrical  or  they  may  be  conical  as  in  the  Grant  bearing  shown 
in  Fig.  97.  The  construction  here  shown  permits  of  adjust- 
ment for  wear,  which  is  difficult  to  obtain  where  the  roller  is 
cylindrical. 

If  the  direction  of  the  load  to  be  carried  is  axial,  roller  thrust 
bearings  of  the  form  shown  in  Fig.  98  are  often  used.     The  shaft 


FIG.  98. 


FIG.  99. 


A  carries  a  thrust  collar  B  and  the  thrust  is  taken  on  the  frame  of 
the  machine  by  a  corresponding  collar  C.  A  hardened  steel 
ring  D  is  attached  to  B  and  rotates  with  it,  while  a  similar  ring 
E  is  fastened  to  the  stationary  part  C.  The  conical  rollers  G 
roll  between  these  rings,  carrying  with  them  the  cage  F.  A  thrust 
ring  H  prevents  the  rollers  from  moving  radially  outward.  The 
apex  angle  of  the  roller  should  not  exceed  15°,  and  in  most  cases 
is  kept  down  to  6°  or  7°  to  prevent  serious  end  pressure  against 
this  retaining  ring.  It  is  evident  that  where  the  roller  is  conical 
in  form,  the  apex  of  the  cone  lying  in  the  centre  line  of  the  shaft, 
the  velocity  of  any  point  in  its  periphery  is  proportional  to  its 
distance  from  the  axis  of  the  shaft  and,  theoretically,  true  rolling 
will  be  obtained. 


276  MACHINE    DESIGN 

Bearings  of  this  character  with  conical  rollers  are  expensive 
to  make  in  an  accurate  manner,  and  a  simpler  form,  as  shown  in 
Fig.  99,  is  sometimes  used.  Here  the  rollers  are  cylindrical  in 
form  and  are  made  in  short  lengths  so  as  to  reduce  relative 
slipping.  The  outer  rollers  rotate  faster  than  the  inner  rollers, 
and  the  lengths  and  arrangement  of  the  rollers  are  such  that 
ridges  are  not  worn  in  the  seat. 

Space  does  not  permit  of  discussion  of  the  many  forms  of 
roller  bearings  on  the  market;  but  their  fundamental  principles 
are  the  same,  and  the  student  is  referred  to  current  trade  cata- 
logues for  variations  in  methods  of  construction. 

no.  Allowable  Bearing  Pressures.     It  is  evident   that   the 
bearing  pressure  in  roller  bearings  must  not  be  great  enough  to 
stress  the  material  of  either  roller  or  bearing  surface  beyond  the 
elastic  limit,  but  theoretical  considerations  are  of  little  service  in 
the  actual  designing  of  such  bearings.     The  most  reliable  ex- 
perimental data  bearing  on  the  subject  are  the  results  of  Stribeck's 
work.     In  roller  bearings  under  radial  pressure  the  load  is  con- 
sidered as  carried  on  one-fifth  of  the  total  number  of  rollers;   and 
the  quantity  equivalent  to  the  projected  area  of  the  ordinary 
bearing,  as  far  as  carrying  capacity  is  concerned,  is  considered 
as  the  product  of  length  and  diameter  of  a  single  roller,  multi- 
plied by  one-fifth  the  total  number  of   rollers  in  the  bearing. 
Thus,  according  to  Stribeck,  for  cylindrical  bearings,  if 
N  =  total  number  of  rollers. 
W  =  total  load  on  bearing  in  Ibs. 
w  =  load  on  one  roller  in  Ibs. 
d  =  diameter  of  roller  in  inches  (mean  diameter  for  conical 

rollers) . 

/  =  length  of  roller  in  inches. 
k  =  a  constant  to  be  determined  experimentally. 

=  kld         .      \l    .    ';••••;•....     .......      (i) 

N 
=  kl  d-    %     .      .      ;......      .     J.      .     (2) 

From  Stribeck' s*  experiments  k  has  a  value  of  550  for  unhardened 
rollers  and  bearin-g  surfaces  and  1,000  for  hardened  surfaces. 

*  See  Transactions  A.  S.  M.  E.,  Vol.  XXVII,  page  444. 


CONSTRAINING    SURFACES 


277 


In  the  case  of  thrust  bearings  the  load  may  be  considered  as 
distributed  over  the  cotal  number  of  rollers.  Bearings  of  the 
type  shown  in  Fig.  99  have  been  constructed  to  carry  a  load  of 
156,000  pounds  at  250  revolutions  per  minute. 

BALL  BEARINGS 

in.  Theoretical  Considerations.  Let  the  ball  A, Fig.  100  (b), 
roll  along  the  circular  path*  B,  with  pure  rolling  motion,  making 
point  contact  with  the  path.  Let  the  path  B  be  parallel  to  the 
plane  CD,  and  suppose  also  that  the  ball  as  it  rolls  remains  a 
fixed  distance  from  this  same  plane.  Then  it  is  evident  that  if 
A  rolls  with  pure  rolling  motion  along  B,  it  will  rotate  around 


FIG.  100  (a). 


FIG.  100  (b). 


some  one  of  its  diameters,  at  right  angles  to  B,  as  an  axis,  and 
will  make  contact  with  B  along  the  edges  of  such  a  disc  as  would 
be  cut  from  it  by  a  plane  passing  through  the  point  of  contact 
b  perpendicular  to  the  diameter  around  which  the  ball  rotates. 
Thus  the  ball  may  rotate  around  Ok  as  an  axis,  and  roll  along 
the  edges  of  the  disc  b  i.  It  is  clear,  however,  that  the  ball  can 
rotate  around  only  one  diameter  at  a  time,  and  preserve  true 
rolling  contact  with  B.  If  the  ball  has  two  concentric  paths  of 
contact  as  B  and  E,  Fig.  100  (b)  whose  points  of  contact  with  the 
ball  are  b  and  e  (Fig.  100  a)  respectively,  then  it  must  roll  along 
two  discs  b-i  and  e-l,  and  these  discs  must  have  a  common  axis 
of  rotation  O  k  perpendicular  to  their  planes  and  passing  through 
the  centre  of  the  ball.  Further,  the  discs  must  be  so  placed  that 

*  The  guiding  surfaces  of  ball  bearings  are  almost  invariably  circular  in  form. 


278  MACHINE    DESIGN 

the  lines  i  I  and  b  e  intersect  on  the  line  o  m,  passing  through  the 
common  centre  of  B  and  E;  for  then 

p  e       r  b 
77  =  £7' 

or  the  circumferences  of  the  rolling  discs  are  proportional  to  the 
circumferences  of  the  paths  of  contact,  and  true  rolling  may  be 
attained.  It  is  not  possible  to  have  more  than  two  points  of 
contact  between  the  ball  and  one  of  its  guiding  surfaces,  with 
pure  rolling,  as  the  proportionality  given  above  is  not  true  for 
any  other  points  on  the  line  o  b  except  those  given.  The  above 
principles  are  fundamental  and  apply  to  all  ball  bearings  with 
circular  guiding  surfaces. 

112.  Spinning.     Usually  one  of  the  guiding  members  is  fixed 
and  the  other  rotates,  the  friction  between  the  moving  member 
and  the  ball  causing  the  latter  to  roll.     If  the  load  carried  is  so 
small  that  no  distortion  of  the  surfaces  takes  place,  and  true  point 
contact  exists,  this  frictional  force  will  act  tangent  to  the  outer 
circumference  of  the  disc  of  contact  and  be  parallel  to  its  plane. 
Such  theoretical  conditions  never  exist  in  practice,  as  the  sur- 
faces of  contact  are  deformed,  even  under  light  loads,  and  the 
load  is  carried  on  a  small  area  instead  of  a  point.      The  fric- 
tional force  rotating   the   ball   is,  hence,  indeterminate  and  in 
general  has  components  which   tend   to  rotate  the   ball   about 
other  axes  than  the  one  which  will  give  pure  rolling   motion. 
It  is  clear  that  inaccurate  workmanship  will  give  the  same  result. 
This  action  is  known  as  spinning  and  is  necessarily  accompanied 
by  friction. 

113.  Forms  of  Bearings.     Ball  bearings  are  divided  into  three 
types,  according  to  the  character  of  the  load  and  the  way  it  is 
sustained  by  the  bearing : 

(a)  Radial  bearings,  for  loads  acting  at  right  angles  to  the  shaft. 

(b)  Thrust  bearings,  for  loads  acting  parallel  to  the  axis  of 

the  shaft. 

(c)  Angular  bearings,   for  taking  loads  both  perpendicular 

and  parallel  to  the  axis  of  the  shaft. 


CONSTRAINING    SURFACES 


279 


Each  of  these  types  may  be  either  a  two-paint,  three-point  or  four- 
point  bearing,  depending  on  the  number  of  points  of  contact  made 
r  y  the  ball  on  the  guiding  surfaces. 

114.  Radial  Bearings.  Figure  101  (a)  shows  a  two-point  radial 
bearing.  The  race  B  is  secured  to  the  shaft  A ,  while  the  race  F 
is  secured  to  the  other  member  C.  Either  A  or  C  may  be  the 
rotating  part.  In  order  to  place  the  balls  in  the  raceway  an 
opening  is  often  cut  in  the  side  of  one  of  the  races,  as  shown  at  E, 
and  the  opening  then  closed  with  a  filling  piece  as  shown.  If  the 
race  F  is  stationary  this  filling  piece  can  be  located  on  the  un- 
loaded side  and  no  wear  brought  upon  it.  If  B  is  stationary  the 
opening  must  be  cut  in  it,  and  the  same  care  used  in  locating  the 


FIG.  10 1  (a). 


FIG.  ioi  (b). 


filling  piece  with  reference  to  the  load.  If  both  the  shaft  A  and 
hub  C  rotate  this  cannot  be  accomplished,  and  the  full  load  is 
brought  upon  this  filling  piece,  thus  decreasing  the  capacity  of  the 
bearing  to  sustain  a  load,  on  account  of  the  break  in  the  surface 
of  the  race.  If  the  velocity  of  the  rotating  member  is  high  this 
break  in  continuity  of  the  race  is  destructive  to  the  bearing. 

If  about  half  the  total  number  of  balls  necessary  completely  to 
fill  the  race  is  used,  each  race  may  be  made  of  one  solid  piece. 
In  such  cases  the  bearing  is  assembled  by  moving  the  inner  race 
over  eccentrically  to  the  outer  race,  filling  in  the  balls  and  then 
distributing  them.  Separators  of  elastic  material  are  then  pushed 
in  between  the  balls  to  maintain  correct  spacing.  These  separat- 
ors, also,  often  act  as  reservoirs  for  lubricant.  They  may  be  of 


280 


MACHINE    DESIGN 


felt  or  such  soft  material  or  may  be  made  in  the  form  of  a  helical 
spring.  This  construction  is  shown  in  Fig.  101  (b).  The  lessened 
number  of  balls  is  compensated  for  by  using  balls  of  larger 
diameter  and  hence  greater  carrying  capacity. 

The  carrying  capacity  of  radial  ball  bearings,  according  to 
Stribeck's  experiments,  is  not  affected  materially  by  velocity, 
within  reasonable  limits,  so  long  as  the  velocity  of  rotation  is 
uniform;  but  sharp  variations  of  velocity  at  high  speed  reduce 
the  capacity. 

115.  Thrust  Bearings.  Fig.  102  illustrates  a  four-point  thrust 
bearing.  Here  there  is  no  difficulty  in  filling  in  the  balls  when  the 
races  are  solid.  In  Fig.  102  the  angles  </>  and  </>'  are -equal,  but 
this  is  not  necessary  as  it  is  evident  that  any  line  drawn  through 


FIG.  102. 


O  and  intersecting  the  ball  circle  will  locate  a  pair  of  rolling  discs 
which  will  roll  on  B,  without  interfering  with  the  pair  shown 
which  may  roll  on  A. 

The  surfaces  C  and  D  are  sometimes  made  both  flat  and 
parallel.  It  is  difficult,  however,  to  obtain  absolute  parallelism, 
initially,  between  C,  D  and  the  ball  races,  and  much  more  difficult 
to  maintain  this  parallelism  under  running  conditions.  An  error 
in  alignment,  either  from  poor  workmanship  or  deflection  under 
load,  of  less  than  one  thousandth  of  an  inch  will  cause  concen- 
trated loading  of  the  balls  on  one  side.  If  possible,  therefore, 
such  bearings  should  be  seated  on  spherical  surfaces,  as  shown 
at  Z>,  thus  allowing  the  races  to  adjust  themselves  correctly. 


CONSTRAINING    SURFACES 


28l 


Mr.  Henry  Hess  states  that  speed  is  an  important  factor  in  such 
bearings  and  he  gives  1,500  revolutions  per  minute  as  a  max- 
imum. 

A  simple  form  of  ball  thrust  bearing  is  shown  in  Fig.  103. 
Here  the  balls  run  against  flat  hardened  surfaces,  A  and  B,  and 
are  kept  in  position  by  a  cage  C  made  of  some  soft  alloy.  The 
cage  may  be  made  to  retain  the  ball  loosely  by  drilling  the  open- 
ings for  the  balls  almost  through  as  shown  in  Fig.  103  (b),  in- 
serting the  ball  and  then  closing  down  the  upper  edge  a  little  with 
a  set  as  shown  at  et  Fig.  103  (b). 

116.  Angular  Bearings.  If  possible,  radial  loads  should  be 
supported  by  radial  bearings,  axial  loads  by  thrust  bearings, 
and  angular  bearings  should  be  avoided.  Radial  bearings  should 


(d) 


not  sustain  heavy  axial  loads  and  thrust  bearings  should  not  be 
loaded  axially.  For  light  loads  the  angular  bearing  will  sustain 
pressure  in  either  of  these  directions.  There  are  innumerable 
forms  of  angular  bearings.  Fig.  104  (a),  (b),  (c),  and  (d)  may  be 
taken  as  typical  of  two-,  three-,  and  four-point  angular  bearings. 
The  races  can  be  made  continuous  in  all  cases,  and  are  often  ad- 
justable. This  last  feature,  while  sometimes  necessary  and  often 
claimed  to  be  an  advantage,  is  really  a  detriment  as  it  puts  the 
bearing  at  the  mercy  of  an  unskilled  person.  Properly  designed 
ball  bearings  do  not  wear  appreciably,  and  if  wear  does  take 
place  it  will  occur  on  the  loaded  side  only;  and  adjustment  cannot 
compensate  for  this,  but  only  hastens  the  failure  of  the  bearing. 
It  is  evident  that  all  the  arrangements  shown  in  Fig.  104 


282  MACHINE    DESIGN 

fulfill  the  requirements  for  pure  rolling  contact  as  outlined  in 
Art.  109.  The  path  of  the  ball  is  not  so  definitely  determined 
at  a,  Fig.  104,  as  in  the  other  forms.  For  this  reason  the  radius 
of  the  ball  races  should,  in  order  to  prevent  wedging  of  the  ball, 
not  be  greater  than  three-quarters  the  diameter  of  the  ball.  For 
the  same  reason  the  angle  (f>  in  Fig.  104  (b)  should  not  be  less  than 
about  25°.  In  Fig.  104  (b)  and  104  (c)  the  point  a  may,  theoreti- 
cally, be  anywhere,  as  long  as  it  lies  between  the  discs  which  roll 
on  the  outer  raceway.  It  should  be  so  placed,  however,  as 
nearly  to  equalize  the  loads  at  b  and  c. 

117.  Allowable  Load.     The   allowable  load   which   may  be 
put  upon  a  ball  bearing  will  depend  on  the  following : 

(a)  The  character  of  the  materials  forming  the  balls  and  races. 

(b)  The  shape  of  the  raceways. 

(c)  The  diameter  of  the  balls. 

(a)  Ball  bearings  fail  by  overstressing  the  material  of  the 
raceways  or  balls.     If  the  stress  induced  is  far  beyond  the  elastic 
limit,  and  often  repeated,  the  surfaces  will  flake  off  and  failure 
will  occur.     Experiments  on  the  crushing  strength  of  balls  or 
races  are  useless  and  misleading  as  the  life  of  the  bearing  depends 
on  the  elastic  and  not  the  crushing  strength.     Evidently  none 
but  hard  materials  can  be  used  for  appreciable  loads  and  these 
must   be   homogeneous   in    texture.      Case-hardened   materials 
are  of  doubtful  value  for  severe  service.     For  most  trying  cir- 
cumstances special  steels  and  alloys  will  no  doubt  be  much  used. 

(b)  Theoretically,  a  ball  supports  the  load  on  a  point,  but 
practically  the  unavoidable  distortion  of  the  material  increases 
the  point  to  a  small  surface.     It  can  be  demonstrated  mathe- 
matically, and  is  evident  on  reflection,  that  a  greater  bearing 
surface  will  be  formed  for  a  given  distortion  of  ball  and  ball  race 
the  more  closely  the  cross-section  of  the  ball  race  corresponds  to 
the  cross-section  of  the  ball.     On  the  other  hand,  and  as  a  di- 
rect consequence  of  this  increase  of  surface,  it  is  found  that  the 
friction  increases  as  the  cross-section  of  the  races  approaches  the 
cross-section  of  the  ball,  a  result  to  be  expected. 

It  is  almost  impossible  to  machine  and  adjust  ball  bearings 
of  three-  or  four-point  contact  so  that  the  load  is  uniformly  dis- 


CONSTRAINING    SURFACES  283 

tributed  at  the  various  points  of  contact.  It  is  borne  out  by 
experiment  and  it  is  well  known  that  two-point  bearings  can  carry 
heavier  loads,  than  any  other  form  for  a  given  diameter  of  ball. 

(c)  The  allowable  load  which  a  ball  can  carry  varies  with  the 
square  of  the  diameter. 

These  statements  have  been  proven  experimentally  by 
Stribeck,  who  found  that  the  carrying  capacity  of  a  ball  could  be 
expressed  by 

w  =  k  d2  .      . (i) 

where  w  =  greatest  load  on  one  ball  in  pounds. 

k  =  a  constant  depending  on  the  material  and  shape  of 

ball  races. 
d  =  diameter  of  ball  in  inches. 

Stribeck  assumes  that  the  total  load  is  carried  on  one-fifth  of  the 
total  number  of  balls.  If,  therefore,  W  be  the  total  load  in 
pounds  on  one  row  of  balls,  and  N  the  total  number  of  balls, 

W  =  w—  =  k  d*—  (2) 

5  5 

For  hardened  steel  races  made  of  good  quality  of  steel 

k  =  450  to  750  for  flat  or  conical  races,  three-  or  four- 
point  contact. 

k  =1,500  for  two-point  contact   and  raceways  whose 
radius  of  curvature  equals  %  d. 

With  more  perfect  materials  Stribeck  states  that  these  values 
may  be  increased  fifty  per  cent. 

118.  Practical  Considerations.  It  is  clear  that  in  order  to 
insure  an  even  distribution  of  load,  initially,  the  workmanship 
on  both  balls  and  races  must  be  very  accurate;  and  in  order  to 
maintain  this  distribution  the  material  must  be  uniform  in  quality 
and  hardness  throughout.  It  is  also  found  that,  for  best  results, 
the  surfaces  must  be  highly  polished  and  free  from  scratches. 
The  bearing  must  be  kept  free  from  acid  and  rust  and  provision 
made  for  excluding  dust  and  grit  and  for  retaining  a  supply  of 
lubricant,  the  function  of  the  lubricant  being  largely  to  prevent 
rusting. 


284  MACHINE    DESIGN 

As  before  stated,  it  has  been  found  better  to  carry  the  load  on 
one  row  of  balls,  if  possible.  Where  this  cannot  be  done  special 
provision  should  be  made  to  insure  that  each  of  the  several  rows 
of  balls  carry  its  proportionate  load.  This  usually  leads  to 
some  form  of  equalizing-device  which  complicates  the  design. 

The  minimum  diameter  of  the  shaft  is  fixed,  approximately, 
by  the  load  carried,  and  balls  are  made,  commercially,  in  stand- 
ard sizes.  In  designing  a  bearing  for  a  full  number  of  balls  a 
tentative  computation  must  generally  be  made  to  fix  the  proper 
number  and  diameter  of  the  balls.  Knowing  these,  the  exact 
diameter  of  the  circle  passing  through  the  centre  of  the  balls  can 
be  determined  as  follows : 

Referring  to  Fig.  101  (a),  draw  a  line  connecting  the  centres  of 
any  two  balls  in  contact  as  G  and  H,  and  draw  the  radii  O  G, 
O  H  and  O  i,  as  shown.  Also  let  r  be  the  radius  of  the  ball  and  R 
the  radius  of  the  ball  circle. 

Then  r  =  R  sin  d 

180° 
and   0=-ff- 

.'.  R  = 


180 


This  value  must  be  increased  sufficiently  to  allow  for  the  necessary 
clearance. 

The  methods  of  applying  ball  bearings  are  so  numerous  and 
varied,  that  no  attempt  can  be  made  here  to  illustrate  them,  and 
the  student  is  referred  to  the  following  sources  of  information  on 
this  point  : 

Transactions  of  A.  S.  M.  E.,  Vol.  XXVII  and  Vol.  XXVIII. 

Trade  publications  generally. 


CHAPTER  XI 
AXLES,  SHAFTS,  AND  SHAFT    COUPLINGS 

Hg.  General.  The  terms  axle,  shaft,  and  spindle  are  applied 
somewhat  indiscriminately  to  machine  members  which  are  so 
constrained  by  journals  and  bearings  as  to  admit  of  motion  of 
rotation.  These  rotating  members  may  be  subjected  to  simple 
torsion  or  bending,  or  to  combinations  of  torsion  and  bending. 
Shear,  also,  usually  exists  as  in  the  case  of  loaded  beams. 
Rotating  members  may  be  classified  roughly  as  follows,  according 
to  the  predominating  stress  (see  Art.  26),  or  to  the  particular 
purpose  for  which  they  are  intended.  ' 

(a)  Axles,  loaded  transversely  and  subjected  principally  to 
bending. 

(b)  Shafts,   subjected  to  torsion  or  combined  torsion  and 
bending. 

(c)  Spindles,  or  short  shafts  which  directly  carry  a  tool  for 
actually   doing  work,  and  which  as  a  consequence  must  have 
accurate  motion. 

The  axles  of  railway  freight  cars  are  good  examples  of  case 
(a);  transmission  shafting  in  factories,  or  the  shafts  of  steam 
engines  are  good  examples  of  (b) ;  while  lathe  and  milling-machine 
spindles  illustrate  (c). 

Considerations  of  strength  seldom  enter  into  the  design  of 
spindles.  In  these  members  torsional  stiffness  and  accuracy  of 
form  in  the  bearings  are,  usually,  the  most  important  considera- 
tions. When  the  spindle  is  designed  with  these  latter  require- 
ments in  view,  there  is  usually  an  excess  of  strength  against 
rupture.  The  discussions  given  in  Art.  12  apply  in  this  case, 
and  it  will  not  be  considered  further  here. 

120.  Axles.  Let  A  (Fig.  105)  be  an  axle  which  carries  the 
loads  Pj,  P2  and  P3,  but  is  not  subjected  to  any  torsional  stress 

285 


286 


MACHINE    DESIGN 


except  that  due  to  negligible  bearing  friction.  Suppose  the  axle 
to  be  supported  by  the  bearings  N  and  N.  The  distribution  of 
the  bearing  reactions  is  indeterminate,  as  explained  in  Art.  95, 
and  the  assumption  is  usually  made  that  they  are  concentrated  at 
the  middle  of  the  bearings,  as  indicated.  This  assumption  is 
on  the  safe  side,  so  far  as  the  strength  of  the  shaft  is  concerned,  as 
the  slightest  deflection  of  the  shaft  tends  to  concentrate  the  re- 
action at  the  inner  edge  of  the  bearing.  The  axle  can,  therefore, 
be  treated  as  a  simple  beam  (Art.  14).  If  the  load  P2  were  zero, 
and  the  loads  Pl  and  P3  were  equal  and  symmetrically  placed 
(which  is  the  most  usual  condition,  as  in  car  axles),  the  case 


SPACE  DIAGRAM 

FIG.  105. 


would  be  identical  with  Case  XIV  of  Table  I.  It  will  be  in- 
structive, however,  to  make  a  solution  of  the  general  case  given 
above. 

The  principal  stress  to  which  the  axle  is  subjected  is  simple 
bending.  Shear  also  exists  in  every  section;  but  from  the  general 
theory  of  beams  (Art.  14)  it  is  known  that,  usually,  this  latter  may 
be  neglected  in  the  body  of  the  shaft.  If,  however,  the  shaft  is 
short,  and  consequently  need  not  be  large  to  withstand  the  applied 
bending  moment,  the  section  of  the  bearing  at  XX  should  be 
checked  for  shearing  stress.  The  dangerous  section  of  the  shaft 
will  be  where  the  bending  moment  is  a  maximum,  and  hence 
it  is  necessary  to  determine  this  maximum  moment,  which  also 


AXLES,    SHAFTS,    AND    SHAFT   COUPLINGS  287 

involves  the  determination  of  the  unknown  reactions.  The 
reactions  may  be  determined  mathematically  by  taking  moments 
around  R2. 

Then,  RJ  =  P,l,  +  P212  +  P313 


and  R    =  P    +  P2  +  P3  - 


The  bending  moment  at  any  section  is  the  algebraic  sum  of  all 
the  moments  at  either  side  of  the  plane  considered.  Thus  the 
bending  moment  at  P2  =  M2  =  Rl  (I  -  12)  -  Pl  (lt  -  12)  and 

this  value  may  be  used  in  equation  J  of  Table  VI  \M2  =  —j 

to  determine  the  stress  for  a  given  cross  section,  or  to  determine 
the  cross  section  for  an  assumed  stress. 

A  graphical  solution  is  much  more  convenient,  as  it  shows 
at  once  where  the  maximum  bending  moment  is  located.  In 
Fig.  105,  denote  the  forces  Plt  P2,  etc.,  thus,  ab,  be,  cd,  etc.,  and 
draw  the  corresponding  force  diagram  as  shown,  making  AB  = 
Plt  BC  =  P2,  and  CD  =  P3,  to  any  convenient  scale.  It  is  to 
be  noted  that  these  forces  are  drawn  consecutively  downward, 
since  they  act  in  that  direction,  and  their  sum,  AD,  must  equal 
the  sum  of  the  reactions,  or  vertical  forces.  Take  any  convenient 
pole,  as  O,  and  draw  OA,  OB,  OC  and  OD.  From  any  point  on 
ab,  in  the  space  diagram,  draw  oa  and  ob,  parallel  respectively  to 
OA  and  OB  in  the  force  diagram.  From  the  intersection  of  ob 
and  be  draw  oc,  parallel  to  OC,  and  in  similar  manner  draw  od. 
Join  the  intersection  of  oa  and  ea  with  the  intersection  of  od  and 
de,  thus  locating  the  closing  string  oe.  Draw  OE  parallel  to  oe, 
locating  E.  Then  in  the  force  diagram  DE  =  R2,  and  EA  =  /?, 
to  the  assumed  scale  of  the  force  diagram. 

The  vertical  ordinates  of  the  space  polygon  are  proportional 
to  the  bending  moments  at  the  points  considered.  The  numerical 
value  of  any  bending  moment  is  the  continued  product  of  the 
length  of  the  ordinate,  the  perpendicular  distance  of  O  from  AD, 
the  reciprocal  of  the  scale  of  the  space  diagram,  and  the  reciprocal 


288  MACHINE    DESIGN 

of  the  scale  of  the  force  diagram.  Thus  if  the  ordinate  at  some 
point  be  2"  long,  the  pole  distance  be  2X">  the  space  scale  be 
iX"  to  i  ft.,  or  y%  size,  and  i"  =  5,000  Ibs.  on  the  force  diagram; 
then  the  bending  moment  at  the  point  considered  is  M  =  2  X 
2>^  X  &  X  5,000  =  200,000  inch  Ibs.;  and  from  this  moment 
the  diameter  of  the  shaft  may  be  computed. 

121.  Shafts  Subjected  Principally  to  Torsion.  The  funda- 
mental relations  existing  in  a  shaft  which  is  subjected  to  torsion 
only  have  been  fully  discussed  in  Article  12,  and  for  such  cases 
or  where  other  stresses,  such  as  those  due  to  bending,  are  negli- 
gible, Article  12  is  applicable.  Shafts  subjected  to  pure  torsion 
rarely  occur  in  practice,  as  bending  is  almost  always  present 
due  to  the  weight  of  the  shaft  itself,  and  to  the  weight  of  pulleys 
which  it  supports,  as  well  as  to  belt  pull,  etc.  There  are  many 
cases,  however,  where  the  torsional  stress  is  predominant,  and 
where  the  secondary  bending  effect  is  difficult  to  compute.  Thus 
in  long  factory  shafting,  where  the  power  is  supplied  to  the  shaft 
at  one  point,  and  is  given  off  in  small  increments  at  short  inter- 
vals all  along  the  shaft,  the  bending  due  to  the  pull  of  the  belts 
is  small.  This  is  especially  true  if  care  is  exercised  to  place  the 
pulleys  as  close  to  the  bearings  as  possible. 

If  the  shaft  is  of  considerable  length,  the  angular  distortion 
is  of  importance,  and  it  may  often  occur  that  a  shaft  having 
sufficient  torsional  strength  will  not  have  proper  torsional  stiffness. 
If  the  power  is  applied  at  one  end  of  the  shaft,  and  taken  off  at 
the  other  end,  computations  for  both  strength  and  stiffness  are 
easily  made  and  may  be  of  service.  In  nearly  all  cases,  how- 
ever, power  is  delivered  in  varying  quantities  all  along  the  shaft, 
and  such  computations  are  not  only  difficult  to  make  but  would 
indicate  that  the  diameter  of  the  shaft  should  vary  at  different 
parts  of  its  length.  This  would  be  undesirable,  as  it  is  important 
that  shafting,  hangers,  etc.,  should,  as  far  as  possible,  be  uniform 
and  interchangeable  for  convenience  and  economy;  and  the 
practice  of  reducing  the  diameter  of  the  shaft  as  it  extends  from 
the  driving  point  is  confined  to  larger  shafting  (say  over  3"  in 
diameter).  The  design  of  shafts  subjected  principally  to  torsion, 
therefore,  is  usually  based  on  the  formula  for  torsional  strength, 


AXLES,    SHAFTS,    AND    SHAFT   COUPLINGS  289 

modified   by   practical  coefficients  which  experience  has  shown 
will  provide  for  stiffness  against  torsion  and  bending. 
Referring  to  equation  E,  Article  12, 


If  P  be  the  equivalent  force  applied  at  the  periphery  of  the  shaft, 
so  that  T  =  Pr,  where  r  is  the  radius  of  the  shaft  in  inches ;  and 
if  N  be  the  number  of  revolutions  of  the  shaft  per  minute;  then 
the  horse  power  transmitted  will  be 

2Prr,N  2T-N 

H.  P.  = 


33,000  X  12       33,000  X  12 

33,000  X  12  X  H.  P.  _  63,024  H.  P. 

2-N  N 

Substituting  this  value  of  T  in  (i)  above, 


where  k  is  a  constant  depending  on  the  stress  assigned.  If  shear- 
ing stress  alone  were  to  be  considered,  ps  might  be  taken  as  high 
as  9,000  Ibs.  per  square  inch,  for  steel  shafting.  In  order  to 
secure  stiffness,  and  to  provide  for  the  indeterminate  bending  in 
line  shafts,  it  is  customary  to  assume  a  lower  stress  (or  higher 
factor  of  safety),  depending  on  the  material  used,  and  the  service 
for  which  the  shaft  is  intended.  The  larger  and  more  important 
the  shaft,  the  lower  should  be  the  working  stress,  as  the  failure 
of  a  head  shaft  or  shaft  of  a  prime  mover  is  accompanied  by  great 
inconvenience  and  expense.  The  following  factors  of  safety  are 
indicated  by  successful  practice:  — 

For  head  shafts  ........     15 

"     line  shafts  carrying  pulleys   .      .      .     10 
"     small  short  shafts,  countershafts,  etc.     7 

For  steel  shafting,  the  allowable  stress  for  the  above  factors  would 
be  about  4,000,  6,000,  and  8,500  respectively,  whence 


2QO  MACHINE    DESIGN 

For  head  shafts, 


For  line  shafting  carrying  pulleys, 


'-3.75^-  (4) 


For  small  short  shafts,  countershafts,  etc., 


It  must,  however,  be  borne  in  mind  that  a  universal  rule 
cannot  be  laid  down  for  any  class  of  shafting;  and  cases  will 
always  arise  which  need  further  consideration  than  given  by  the 
above  equations.  For  example,  in  the  span  of  shafting  where 
the  power  is  applied  by  a  large  belt  the  bending  action  may  be 
excessive,  and  this  particular  span  may  have  to  be  of  larger  diam- 
eter than  the  remainder  of  the  shaft.  The  student  is  referred 
to  handbooks*  for  tabulated  data  on  the  size  of  transmission 
shafts  for  various  purposes.  It  is  to  be  especially  noted  that  a 
shaft  carrying  a  transverse  load,  which  applies  a  bending  moment 
to  the  shaft,  is  subjected  to  a  reversed  stress  as  the  shaft  rotates. 
If,  in  addition,  the  twisting  moment  varies  in  magnitude,  the 
factor  of  safety,  owing  to  complete  or  partial  reversal  of  stress 
(see  Arts.  25  and  26),  must  be  high,  and  this  accounts  for  the 
low  stresses  allowable  with  such  shafts. 

122.  Shafts  Subjected  to  Torsion  and  Bending.  In  engine 
shafts,  head  shafts  driven  by  heavy  belts,  and  many  others,  the 
tdrsional  stress  is  not  predominant  and  may,  in  fact,  be  less  than 
that  due  to  bending.  A  full  discussion  of  the  relations  which 
exist  in  this  case  has  been  given  in  Article  16  and  it  remains  to 
show  the  application  of  this  discussion  to  actual  cases  of  design. 

From  Article  16  (equations  K,  K^  and  Fig.  9),  it  appears 
that  if  the  bending  and  twisting  moments  can  be  determined  for 
any  section,  the  theoretical  diameter  of  the  shaft  at  that  section 

*  See  Kent's  "  Mechanical  Engineer's  Pocket  Book,"  page  869. 


AXLES,    SHAFTS,    AND    SHAFT   COUPLINGS    '  29 1 

can  be  found.  Usually  the  twisting  moment  can  be  determined 
without  difficulty,  but  the  bending  moment  is  often  difficult  to 
determine,  and  sometimes  the  designer  must  be  content  with  an 
approximation.  One  of  the  greatest  sources  of  uncertainty  is 
the  location  of  the  reactions  at  the  bearings.  Usually,  as  already 
pointed  out,  the  safe  assumption  is  made  that  these  reactions  are 
concentrated  at  the  centre  line  of  the  bearing.  When  the  shaft 
is  of  appreciable  length  (15  or  20  diameters),  the  error  is  small; 
but  in  such  cases  as  the  crank  shafts  of  multiple-cylinder  engines, 
where  the  distance  between  the  centres  of  bearings  is  only  four 
or  five  diameters,  or  less,  it  is  evident  thaf  the  assumption  is 
in  the  direction  of  excessive  safety. 

In  line  shafting,  particularly  with  the  usual  swivel  bearings, 
the  error  from  this  source  is  small,  and  at  first  sight  the  conditions 
of  such  shafting  would  appear  to  approximate  those  of  a  con- 
tinuous beam.  While  such  an  assumption  might  be  safely  made 
when  the  shafting  has  been  put  in  perfect  alignment,  it  would  not 
be  safe  as  a  general  principle,  as  perfect  alignment,  even  under 
best  conditions,  is  of  short  duration,  and  bending  stresses  soon 
appear  as  a  result  of  lack  thereof.  It  appears,  therefore,  that, 
in  this  case,  the  safest  procedure  would  be  to  treat  each  span 
as  if  disconnected  at  the  bearing,  when  computing  bending 
moments. 

A.  typical  case  of  combined  twisting  and  bending  is  the  engine 
shaft  shown  in  Fig.  106  (a),  the  data  taken  being  those  of  the 
example  in  Case  (c),  Art.  5.  Here  the  shaft  is  supported  by  the 
bearings  at  the  points  X  and  X',  as  indicated,  and  carries  a  heavy 
generator  spider  at  Y.  The  weight  of  this  spider,  and  that  of 
the  shaft  itself,  with  the  probable  magnetic  pull  which  may  occur 
when  the  shaft  wears  downward  a  little,  is  estimated  at  22,000 
Ibs.  The  maximum  pressure  (P)  on  the  crank  pin,  due  to  the 
steam  pressure,  is  25,000  Ibs.  This  force  is  a  maximum  when 
the  crank  is  about  vertical,  and,  at  that  position,  it  exerts  a  twist- 
ing moment  on  the  shaft  from  the  crank  to  the  point  F*  where 
power  is  delivered,  and  also  a  bending  moment  on  the  shaft  in  a 

*  The  reinforcing  effect  of  the  hub  of  the  spider  is  neglected. 


292 


MACHINE    DESIGN 


Horizontal  direction.     The  weight  of  the  generator,  etc.,  exerts 
a  simple  bending  moment  in  a  downward  direction,  and  at  right 


FIG.  1 06  (a). 


angles  to  that  induced  by  P.     Fig.  106  (b)  shows,  isometrically, 
the  direction  and  point  of  application  of  the  various  forces  and 


AXLES,    SHAFTS,    AND    SHAFT   COUPLINGS  293 

reactions,  and  it  is  required  to  find  the  maximum  equivalent 
bending  moment  on  the  shaft. 

It  was  shown  in  Example  (c) ,  Article  99,  how  a  tentative  solu- 
tion could  be  made  for  the  diameter  and  length  of  the  main  jour- 
nal, thus  fixing  the  distance  of  its  centre  line  from  the  centre  line 
of  the  crank  pin  at  2iX"-  Other  data  fix  the  distance  between 
bearings  as  f  -  9". 

Graphical  Analysis  is  here  very  convenient,  and  the  order  of 
procedure  will  be  as  follows:— 

(a)  Find  the  bending  moment  due  to  the  steam  pressure  P. 

(b)  Find  the  bending  moment  due  to  the  dead  load  W. 

(c)  Combine  these  bending  moments  to  find  the  maximum 
resultant  bending  moment. 

(a)  Consider,  for  convenience,  that  the  force  P,  and  all  the 
reactions  due  to  it,  have  been  rotated  into  the  plane  of  the  paper 
so   that  P  is  represented  as  acting  vertically.     Draw  the  force  * 
diagram,  O'  Bf  Af  C'  for  force  P,  and  the  reactions  due  to  it,  to 
a  convenient  scale,  here  taken  as  8,000  Ibs.  per  inch,  taking  O'- 
on  a  horizontal  line  through  A',  thus  making  the  closing  string 
of  the  space  diagram  also  horizontal,   which  is  convenient  for 
later  work.     Draw  the  space  diagram,  M  N  P  for  force  P,  and 
the  reactions  due  to  it,  as  shown.     The  scale  of  the  space  dia- 
gram is  y^"  to  i  ft.  or  TV  size. 

(b)  In  a  similar  manner  construct  the  force  diagram  ABC  O, 
for   force   Wt   and  the  corresponding  space    diagram  H  T.  J  /, 
for  the  force  W,  making  the  pole  distance  =  A'  O';    taken  here 

as  3".t 

(c)  To  combine  the  bending  moments  at  any  section,  as  Z, 
take  the  intercept  S  T,  on  H  I  J,  and  lay  it  off  as  S'  T'  on 
the    diagram   M-N  P.      The    distance   T   U  is   proportional 
to    the   combined   bending    moments    and  may  be  used   as  an 
ordinate  VV  in   the    diagram    of   combined   bending  moments 
DGFE. 

It  often  occurs  that  the  shaft  carries  a  heavy  flywheel  at  Y, 


*See  also  Article  120. 

f  Reduced  to  one-half  size  in  cut. 


294  MACHINE    DESIGN 

instead  of  a  generator,  and  a  heavy  belt  may  also  run  on  the 
wheel.  It  is  evident  that  the  resultant  force  due  to  the  weight 
of  the  wheel  and  the  pull  of  the  belt,  can  be  determined,  both  in 
magnitude  and  direction.  In  general,  the  direction  of  this  force 
will  not  be  vertical,  but  will  make  an  angle,  <£,  less  than  90°  with 
the  direction  of  the  force  P.  In  such  a  case  the  moments  may  be 
combined  by  the  triangle  of  forces  taking  into  consideration  the 
angle  <j>. 

The  numerical  value  of  any  moment  is  the  continued  product 
of  the  ordinate  which  represents  it,  the  pole  distance,  the  re- 
ciprocal of  the  scale  of  the  space  diagram,  and  the  reciprocal  of 
the  scale  of  the  force  diagram.  Thus  the  maximum  bending 
moment,  which  occurs  at 

F  =  iTy  X  3  X  V5  X  -      -  =  485,400  inch  pounds. 

The  twisting  moment  is  seen  by  inspection  to  be  uniform 
over  the  whole  length  of  the  shaft  which  it  affects.  Its  numerical 
value  is,  as  before,  25,000  X  18  =  450,000  inch  Ibs.;  and  these 
two  moments  may  be  combined,  to  determine  the  safe  diameter 
of  the  main  part  of  the  shaft  according  to  the  methods  of  Arti- 
cle 1 6.  A  graphical  method  will  be  given  later,  which  some- 
what facilitates  the  numerical  work  of  this  computation. 

The  methods  outlined  above  are  clearly  applicable  to  any 
shaft  which  has  not  more  than  two  points  of  support  since  in  such 
cases  the  reactions  can  be  readily  found. 

A  convenient  diagram  is  shown  in  Fig.  107  for  determining 
the  diameter  of  a  shaft,  of  solid  circular  cross-section,  subjected 
to  any  moment,  and  with  any  intensity  of  fibre  stress  from  zero 
to  15,000  Ibs.  per  sq.  inch.  This  diagram  can  be  used  for 
either  simple  bending  or  twisting  moments,  or  for  combined 
bending  and  twisting  actions.  Its  use  in  connection  with  prob- 
lems involving  simple  twisting  moments  will  be  discussed  first. 

If  T  is  the  twisting  moment,  d  the  diameter  of  the  solid 
circular  shaft,  and  p  the  intensity  of  stress  in  the  most  strained 

fibres,  T  --    ~7pd3.     Therefore,  for  a  given  diameter  of  shaft,  T 


Scale  B 
1         2         345678 


10        11       12       13       14        15 


r7 


x 


.345; 


3  4 

Scale  C 


FIG.  107. 


296  MACHINE    DESIGN 

is  directly  proportional  to  p.  Thus,  if  d  =  4",  d3  =  64,  and 
T  ••=  .196  X  &4p  =  12. ^p.  If  p  be  taken  as  10,000,  T  =  125, 700 
inch  Ibs.  In  Fig.  107  if  ordinates  represent  moments  (to  the 
scale  "A,"  of  500  inch  Ibs.  to  each  division);  and  if  abscissas 
represent  intensity  of  stress  (to  the  upper  scale,  "5,"  of  1,000  Ibs. 
per  sq.  inch  to  each  division),  the  point  a  corresponds  to  T  = 
125,700,  p  =  10,000,  d  =  4".  As  the  moment  varies  directly 
as  the  intensity  of  stress,  for  any  given  diameter  of  shaft,  the 
relations  between  corresponding  values  of  T  and  p  (for  a  4" 
shaft)  will  be  represented  by  the  straight  line  through  the  point 
a,  and  the  origin  O.  In  a  similar  manner  straight  lines  through 
the  origin  are  drawn  for  other  shaft  diameters. 

To  determine  the  diameter  of  shaft  for  a  moment  of  90,000 
inch  Ibs.,  with  a  fibre  stress  of  12,000  Ibs.  per  sq.  inch,  pass  along 
the  horizontal  through  the  point  marked  "9"  (or  T  =  90,000) 
on  scale  "A"  to  the  vertical  line  through  the  point  marked  "12" 
(or  p  =  12,000)  on  scale  "B."  The  intersection  of  this  horizon- 
tal and  vertical  (b)  lies  a  little  below  the  diagonal  marked  3.4 
at  its  outer  end;  or  the  shaft  should  be  about  3.37"  (or  3^4") 
diameter  to  give  a  stress  of  12,000  Ibs.  per  sq.  inch. 

The  oblique  line  nearest  to  the  point  located  in  the  last 
example  bears  three  figures,  viz.:  ".732  —  1.58  —  3.4,"  and  the 
other  diagonals  each  bear  three  separate  figures.  The  signifi- 
cance of  these  designations  will  be  explained  by  further  illustra- 
tions. 

If  T  =  yV  of  90,000,   or  9,000  inch  Ibs.,   and  p  =  12,000, 


3.37 


71  I2,OOO  7T  10 

since  d  varies  as  the  cube  root  of  T,  and  when  T  =  90,000,  d  = 

3-37". 

In  a  similar  way,  if  T  =  900,  or  TJ^  of  90,000,  d  =  3.37  -r- 

\/^o~  =  .726". 

To  use  the  diagram  when  T  =  goo,  and  p  =  12,000,  consider 
scale  "A"  as  representing  the  moment  in  100  inch  Ibs.;  pass 
along  the  horizontal  through  9  on  this  scale  to  the  vertical,  through 
12  of  scale  "J5,"  as  before,  to  the  point  "6,"  and  take  the  first 


AXLES,    SHAFTS,    AND    SHAFT   COUPLINGS  297 

figure  borne  by  the  nearest  diagonal  (.732)  as  the  approximate 
diameter  of  the  shaft;  or,  by  interpolation,  find  the  diameter  = 
.726". 

If  T  =  9,000,  p  =  12,000;  consider  scale  "4"  as  represent- 
ing the  moment  in  1,000  inch  Ibs.,  and  read  the  middle  figure  on 
the  nearest  diagonal  (1.58)  as  the  required  approximate  diameter 
of  the  shaft;  or,  by  interpolation,  the  diameter  is  found  to  be 

i.56". 

If  the  moment  is  greater  than  130,000,  the  diagram  is  quite 
as  applicable  as  for  smaller  moments.  Thus  if  T  =  900,000 
and  p  =  12,000,  consider  scale  "Aj"  as  representing  the 
moment  in  100,000  inch  Ibs.  The  horizontal  through  9  of 
scale"  A"  and  the  vertical  through  12  of  scale  "B"  intersect 
at  "6"  as  before.  The  required  diameter  is  about  7.26";  be- 
cause the  diameter  was  found  to  be  about  .726  for  a  moment 
of  900,  and  it  must  be  10  times  as  great  for  a  moment  of  io3  X 
900  =  900,000.  For  p  =  12,000  with  a  moment  of  9,000,000 
inch  Ibs.  (=  io3  X  9,000),  the  diameter  is  io  X  1.57  =  15.7", 
etc.  It  thus  appears  that  the  diagram  covers  all  moments,  with- 
out being  of  such  impracticable  size  as  it  would  be  if  it  were  not 
for  the  peculiar  designation  of  the  oblique  lines  and  the  method 
of  using  scale  "A"  The  diagram  can  also  be  used  for  simple 
bending  moments.  The  expression  for  the  bending  moment  in 
a  shaft  of  solid  circular  section  is 


while  the  expression  for  a  twisting  moment  is,  as  given  above, 


Therefore,  with  a  given  diameter  and  numerically  equal  fibre 
stress,  T  is  numerically  equal  to  2M.  To  determine  d  for  given 
values  of  p  and  M,  multiply  M  by  2  to  get  the  equivalent  T,  and 
with  this  value  of  T,  proceed  as  in  the  former  examples. 

For  finding  the  diameter  appropriate  to  a  combined  bending 
and  twisting  moment,  the  equivalent  twisting  moment, 

r  =  M  +  v  M2  +  r- 


2Qo  MACHINE    DESIGN 

is  to  be  determined ;  see  Art.  1 6,  equation  K4.  This  equivalent  twist- 
ing moment  is  readily  determined  from  the  diagram  by  the  use  of 
scale  "C"  at  the  bottom  of  Fig.  107  and  a  pair  of  dividers,  when 
the  simple  bending  moment  (M)  and  the  simple  twisting  moment 
{T}  are  given.  Example:  Suppose  M  =  30,000;  T  =  40,000; 
and  p  =  13,000.  Consider  scales  "A"  and  "C"  to  measure 
moments  in  10,000  inch  Ibs.  Take  M  at  3  on  scale  "A"  with 
one  point  of  the  dividers,  and  T  at  4  on  scale  "C"  with  the  other 
point  of  the  dividers;  then  the  distance  between  3  on  scale  "A" 
and  4  on  scale  "C"  represents  \/  M2  +  T2.  Swing  the  dividers 
about  the  point  at  3  on  scale  "A"  as  a  centre  until  the  other 
point  reaches  scale  "A"  (at  point  8) ;  then  o.  .8  on  scale  "A"  = 

o 3+3 8  =  M  +  \/  M2  +  T2  =  Te.  With  the  value  of 

Te,  found  in  this  way,  proceed  as  in  case  of  a  simple  twisting 
moment.  The  intersection  of  the  horizontal  through  8  (Te) 
and  the  vertical  through  13  (p)  is  at  point  "c."  Since  the 
moments  correspond  to  units  of  10,000  inch  Ibs.  on  scale  "A," 
the  largest  figures  of  the  diagonals  are  to  be  read  in  determining 
the  diameter.  The  point  "c"  therefore  indicates  a  diameter  of 
between  3.0"  and  3.2";  by  interpolation  the  diameter  is  taken 
as  3.15".  By  computation  the  diameter  is  found  to  be  3.14". 
A  shaft  3  yV  diameter  would  be  proper  for  this  case.  The  use 
of  the  diagram  in  connection  with  equations  K  and  K,  of  Table 
VI  is  obvious  from  the  above. 

The  diagram  of  Fig.  107  is  equally  convenient  for  finding 
the  intensity  of  stress  in  a  given  shaft  under  a  known  moment; 
or  the  moment  on  a  given  shaft  corresponding  to  any  intensity  of 
stress.  Thus,  if  a  J24"  shaft  is  subjected  to  moment  of  1,000,000 
inch  Ibs.,  consider  the  moment  units  as  100,000  inch  Ibs.,  pass 
horizontally  from  10  on  scale  "A"  to  a  point  slightly  below  the 
diagonal  marked  .776  (7.76"  diameter),  and  then  vertically  up- 
ward to  scale  "By"  where  the  stress  is  read  as  about  11,000  Ibs. 
per  sq.  inch. 

If  it  is  required  to  find  the  twisting  moment  corresponding 
to  an  intensity  of  stress  of  9,000  Ibs.  per  sq.  inch  on  a  shaft  iX" 
diameter;  pass  vertically  downward  from  "9"  on  scale  "B"  to 
a  point  slightly  above  the  diagonal  marked  "1.49";  then 


AXLES,    SHAFTS,    AND    SHAFT   COUPLINGS  2QQ 

horizontally  to  5.9  on  scale  "A"  As  1.49  is  the  middle  number 
on  the  diagonal,  the  moment  units  are  1,000  inch  Ibs.;  therefore 
T  =  5.9  X  1,000  =  5,900  inch  Ibs. 

123.  Torsional  Stiffness  and  Deflection  of  Shafting.  When  a 
shaft  has  considerable  length,  the  matter  of  torsional  stiffness 
is  important.  A  rule,  common  in  practice,  is  to  limit  the  twist 
in  the  shaft  to  one  degree  for  every  20  diameters  in  length. 
Another  rule  limits  the  twisting  to  0.075  degree  for  every  foot 
in  length.  The  lateral  deflection  of  the  shaft  should  not  exceed 
y-JV  per  foot  of  length,  to  insure  proper  contact  at  the  bearings. 
Theoretical  considerations,  however,  do  not  enter  so  largely  into 
the  spacing  of  bearings  of  line  shafting,  as  does  the  construction 
of  the  framework  to  which  the  bearings  are  fastened.  Care 
should  be  exercised  in  laying  out  such  structures,  that  provision 
is  made  for  fastening  the  hangers  close  enough  together  to  avoid 
excessive  deflection.  For  the  average  range  of  velocities  found 
in  practice  the  following  formulae*  can  be  used  for  ordinary 
small  shafting. 

L  =  7  \/  d?  for  shaft  without  pulleys  (i) 

L  =  5  v'd2  f°r  shaft  carrying  pulleys  ...  (2) 
where  L  =  distance  between  hangers  in  feet  and  d  =  diameter 
of  shaft  in  inches. 

If  T  be  the  twisting  moment  in  foot  Ibs.  applied  to  a  shaft, 
then  the  power  transmitted  at  N  revolutions  per  minute  iszTnN; 
from  which  it  appears  that  the  greater  the  velocity  of  the  shaft, 
the  smaller  is  the  required  turning  moment,  for  a  given  amount 
of  power  transmitted. 

If  a  slightly  deflected  shaft  is  rotated,  centrifugal  force,  acting 
on  the  eccentric  mass  of  the  shaft,  tends  to  equalize  the  forces 
which  hold  the  shaft  deflected  in  one  plane  and  to  whirl  the  shaft 
as  a  whole  around  the  axis  of  rotation.  At  low  speeds  the  action 
of  centrifugal  force  is  small,  and  the  deflecting  force  will  hold  the 
shaft  deflected  in  its  plane.  As  the  effect  of  centrifugal  force 
increases  with  the  velocity,  while  the  effect  of  the  deflecting  force 
is  constant,  it  is  clear  that  as  the  speed  is  increased  the  centri- 

*  See  also  Kent's  "Mechanical  Engineer's  Pocket  Book,"  page  869. 


300  MACHINE    DESIGN 

fugal  force  will,  at  some  speed,  balance  the  effect  of  the  de- 
flecting force,  and  the  shaft  will  become  unstable.  Beyond  this 
speed  the  shaft  will  whirl  about  the  central  axis.  For  a  given 
diameter  of  shaft  there  is  one  definite  speed  within  which  it  will 
maintain  a  stable  condition  with  a  given  deflection. 

If  L  =  distance  between  bearings  in  feet,  d  =  diameter  of 
shaft  in  inches,  and  N  =  the  revolutions  per  minute,  then  for  the 
critical  speed  * 


This  equation  refers  to  the  bare  shaft  only  and  it  determines 
the  maximum  safe  span.  Where  pulleys  are  carried  at  some 
distance  from  the  bearings,  the  span,  L,  must  be  less  than  the 
value  given  by  equation  (3)  on  account  of  the  added  mass  of  the 
pulleys,  and  the  great  liability  of  the  latter  to  be  unbalanced. 
The  speed  of  shafting  in  practice  is,  almost  always,  considerably 
below  the  critical  speed. 

124.  Practical  Considerations,  Hollow  Shafting,  etc.  Shafting 
up  to  3"  in  diameter  is,  in  this  country,  made  of  cold-rolled  steel. 
Such  shafting  is  true  and  straight  and  needs  no  turning  whatever. 
If  keyways  are  cut  the  shaft  must,  in  general,  be  carefully  straight- 
ened afterward,  as  the  cutting  relieves,  locally,  the  skin  tension 
due  to  the  cold-rolling  |  thus  causing  the  shaft  to  warp.  Larger 
sizes  of  shafting  are  forged  and  machined. 

The  use  of  hollow  shafts  not  only  reduces  the  weight  for  a 
given  strength,  but  the  removal  of  the  metal  from  the  core  of  a 
steel  shaft  (or  of  the  ingot  from  which  it  is  made)  very  greatly 
increases  its  reliability  under  repeated  application  of  stress. 

Shortly  after  a  steel  ingot  is  cast,  the  exterior  solidifies  and 
becomes  comparatively  cool  while  the  internal  portion  is  still 
fluid.  The  subsequent  contraction,  during  complete  cooling,  is 
much  less  in  the  exterior  walls  than  it  is  in  the  hotter  interior 
mass.  Unless  the  interior  is  "fed"  during  this  period,  it  will  be 
less  dense  than  the  outer  portions  and  shrinkage  cavities  are  apt  to 

*  See  Rankine's  "Millwork,"  page  549. 
f  See  Article  12. 


AXLES,    SHAFTS,    AND    SHAFT   COUPLINGS  301 

be  present  near  the  centre  of  the  ingot.  Numerous  expedients 
have  been  adopted  to  reduce  this  evil,  among  which  is  "fluid 
compression,"  or  subjecting  the  ingot  to  heavy  pressure  immedi- 
ately after  it  is  poured.  The  difficulty  is  not  entirely  overcome 
by  such  means,  however,  as  the  walls  of  large  ingots  become  too 
rigid  to  yield  to  the  pressure  before  the  interior  is  entirely  solidified. 
The  external  walls  "freeze,"  after  which  the  internal  shrinkage  is 
made  up  by  metal  flowing  from  the  upper  portion  toward  the 
bottom  as  long  as  any  of  it  remains  fluid.  This  leaves  a  shrinkage 
cavity  at  the  upper  end  of  the  ingot.  Gas  liberated  during  cooling 
collects  in  this  cavity  also.  The  result  of  these  two  actions  is 
to  form  what  is  called  the  "pipe,"  which  frequently  extends  to  a 
considerable  depth.  The  top  end  of  the  ingot  is  cut  off  and 
remelted,  but  this  does  not  insure  removal  of  all  of  the  pipe,  and 
it  also  involves  much  expense.  If  the  portion  cut  off  is  not 
sufficient  to  remove  all  of  the  pipe,  a  piece  rolled  or  forged  from 
the  ingot  contains  a  flaw  near  the  centre  which  is  drawn  out 
into  a  long  crack  if  the  ingot  is  worked  into  a  long  piece.  The 
rolling  and  forging  may  squeeze  the  sides  of  the  cavity  together 
so  that  it  is  not  easily  detected  at  any  section,  but  as  this  work 
is  done  at  a  temperature  much  below  that  corresponding  to 
welding,  the  defect  is  not  removed.  This  flaw  is  more  or  less 
irregular  or  ragged;  hence  its  form  is  favorable  to  starting  a 
fracture,  under  variations  of  stress,  which  may  finally  extend 
far  enough  to  cause  rupture. 

If  the  ingot  is  bored  out,  the  pipe  is  effectually  removed,  and 
the  metal  remaining  is  superior  to  that  of  a  solid  shaft.  It  will 
be  evident  that  casting  a  hollow  ingot  is  not  the  equivalent  of 
boring  out  one  which  was  cast  solid ;  for  if  the  ingot  is  cast  hollow 
the  outer  and  inner  walls  cool  before  the  intermediate  mass  does, 
and  the  shrinkage  effect  takes  place  in  the  latter.  In  fact,  a 
shaft  made  from  a  hollow  ingot  is  worse  than  the  solid  shaft,  in 
the  respect  that  the  former  has  the  defective  material  nearer  the 
outer  fibres  where  the  stress  is  greater. 


302 


MACHINE    DESIGN 


COUPLINGS  AND   CLUTCHES 

125.  General  Description.  Couplings  are  machine  members 
which  fasten  together  the  ends  of  two  shafts,  so  that  rotary  motion 
of  one  causes  rotary  motion  of  the  other.  Where  the  connection 
is  to  be  broken  only  at  rare  intervals,  as  in  making  of  repairs,  the 
couplings  are  generally  constructed  so  that  they  must  be  partially 
or  wholly  dismantled  to  separate  the  shafts.  Such  couplings  are 
known  as  permanent  couplings.  When  it  is  desired  to  disengage 
the  shafts  at  will,  the  coupling  is  of  a  different  construction  and 
is  generally  known  as  a  clutch.*  The  use  of  clutches  is  not,  how- 


CD 


D 


FIG.  1 08. 


FIG.  109. 


ever,  confined  to  securing  together  the  ends  of  shafting,  but  they 
are  much  used  for  engaging  and  disengaging  pulleys  at  will,  in 
connection  with  the  shafts  on  which  they  are  placed.  For  this 
service  clutches  making  use  of  friction  are  much  used,  and  this 
particular  type  is  discussed  in  Chapter  XIII. 

Couplings  should  be  placed  near  a  bearing,  so  as  to  bring 
the  joint  in  the  shaft  near  a  supported  point,  and  should  be  placed 
on  the  side  of  the  bearing  farthest  away  from  the  point  where 
power  is  applied,  so  that  when  the  shaft  is  disconnected  the 
running  part  is  supported  near  the  end. 

126.  Permanent  Couplings.  Where  the  axes  of  the  two  shafts 
to  be  connected  are  parallel  and  coincident,  couplings  such  as  are 
shown  in  Figures  108,  109,  no,  and  in  are  used.  Fig.  108 
illustrates  a  type  of  coupling  known  as  a  split-muff  coupling. 


*  See  Transactions  A.  S.  M.  E.,  1908,  for  a  full  description  and  discussion  of 
various  forms  of  clutches. 


AXLES,    SHAFTS,    AND    SHAFT   COUPLINGS 


303 


The  parts  A  and  B  are  separated  by  a  small  space  and  can,  there- 
fore, be  clamped  to  the  shaft  by  the  bolts  C.  For  heavy  work  a 
key  as  shown  is  provided,  but  in  lighter  shafting  friction  alone 
may  suffice  to  prevent  relative  rotation. 

Fig.  no  shows  the  Sellers  Muff  Coupling.  Here  the  circular 
tapered  wedges  B,  B,  are  drawn  inward  by  the  bolts  C.  The 
wedges  are  split  as  shown  at  D,  hence  the  tighter  they  are  drawn 
inward  the  more  firmly  they  clasp  the  shaft.  For  light  work  no 
key  is  necessary,  but  for  the  full  capacity  of  the  shaft  keys  are 
advisable. 

Couplings  such  as  shown  in  Figures  108  and  no  are  regularly 


FIG.  no. 


FIG.  in. 


manufactured  in  standard  sizes,  and  the  student  is  referred  to 
the  trade  catalogues  of  manufacturers  for  dimensions  and  capa- 
cities of  such  couplings. 

The  Flange  Coupling,  Fig.  109,  is  one  of  the  most  common 
and  also  one  of  the  most  effective  forms  of  permanent  couplings. 
The  general  proportions  are  usually  designed  empirically,  but  the 
bolts  should  be  designed  so  that  their  combined  resistance  to  a 
torsional  moment,  around  the  axis  of  the  shaft,  will  be  at  least  as 
great  as  the  torsional  strength  of  the  shaft  itself;  and  the  bolts 
should  be  accurately  fitted  so  as  to  distribute  the  load  evenly 
among  them. 

Let  D  =  diameter  of  the  shaft  in  inches 
d  =  diameter  of  the  bolt  in  inches 
n  =  the  number  of  bolts 
r  =  radius  of  bolt  circle  in  inches 
ps  =  allowable  shearing  stress  per  square  inch,  for  steel. 


3°4 


MACHINE    DESIGN 

Then  -    -  p  =  -  -  n  > 
16  4 


whence  d  =  .  q  A   — 


(i) 


Good  practice  gives  n  =  3  +  — ,  but  this  number  may  be 

modified  for  convenience  in  spacing,  etc.  The  bolts  should  be 
carefully  fitted  to  insure  that  each  one  carries  its  full  share  of  the 
load.  The  projecting  outer  flange  is  an  important  feature  as  it 
covers  the  revolving  bolt  heads,  thus  protecting  workmen  from 
becoming  entangled.  For  best  results  the  flanges  should  be 
pressed  on  to  the  shaft  and  the  faces  trued  up  in  place,  thus 


FIG.  112. 

insuring  greater  accuracy  of  alignment.  This  should  be  done 
in  all  good  work. 

When  great  strength  and  reliability  are  desired,  as  in  marine 
work,  the  flange  is  sometimes  forged  solid  with  the  shaft,  as  in 
Fig.  in.  Here  the  bolt  holes  are  sometimes  bored  tapering, 
and  reamed  after  the  flanges  are  placed  together,  thus  insuring 
a  perfect  fit  for  the  bolts,  and  also  facilitating  their  withdrawal. 

When  the  axes  of  the  two  shafts  are  parallel,  but  not  coin- 
cident, or  when  there  is  danger  of  parallel  and  coincident  axes 
wearing  out  of  coincidence,  Oldham's  Coupling,  Fig.  112,  is  often 
used.  It  consists  of  two  heavy  flanges  (A  and  B),  each  keyed  fast 
to  its  own  respective  shaft,  and  an  intermediate  disc  C.  The  disc 
has  a  tongue  running  diametrically  across  each  face,  these  tongues 


AXLES,    SHAFTS,    AND    SHAFT   COUPLINGS 


305 


being  placed  at  right  angles  to  each  other  and  fitting  into  grooves 
cut  in  the  flanges.  With  this  coupling  the  rate  of  rotation  of 
the  driven  shaft  is  identical  with  that  of  the  driver,  or,  in  other 
words,  the  angular  velocity  is  the  same.  The  coupling  is  often 
used  on  the  propeller  shafts  of  small  power  boats. 

If  the  axes  of  the  two  shafts,  A  and  B,  Fig.  113,  intersect  and 
make  an  angle  0  with  each  other  they  may  be  coupled  together 
by  means  of  a  Hook's  Coupling  or  Universal  Joint,  as  it  is  often 
called.  In  this  coupling  each  shaft,  is  fitted  with  a  jaw  D  which 
is  pin-connected  to  an  intermediate  member  F.  The  holes  in 
this  intermediate  member  for  receiving  the  pins  G  are  at  right 
angles  to  each  other.  With  this  arrangement  the  angular  velo- 
city of  the  driven  shaft  is  not  the  same  at  all  points  of  the  revolu- 
tion as  that  of  the  driver.*  The  construction  shown  in  Fig.  113 


FIG.  113. 

is  very  common,  but  the  difference  between  the  angular  velocity 
of  the  driver  and  that  of  the  driven  shaft  is  less  when  the  con- 
struction is  such  that  the  axes  of  the  pins  G  intersect.  The 
construction  required  to  make  the  axes  of  the  pins  intersect  is 
usually  more  complicated  than  that  shown  in  Fig.  113,  and  hence 
in  rough  work  the  simpler  design  is  adopted. 

If  another  shaft  C  be  coupled  to  B  so  that  A  and  C  make  the 
same  angle  0  with  B;  if  also  the  pins  G,  G  in  B  are  parallel  to  each 
other  and  all  three  shafts  lie  in  the  same  plane;  then  the  angular 
velocity  of  C  will  be  identical  with  that  of  A  and  vice  versa. 
Empirical  practice  makes  the  diameter  of  the  pin  G  equal  to 
one-half  the  diameter  of  the  shaft. 

128.  Positive  Clutches.     Positive  clutches  are  much  used  for 


*  See  "Kinematics  of  Machinery,"  J.  H.  Barr,  page  198. 


20 


306 


MACHINE    DESIGN 


starting  and  stopping  such  machines  as  punch  presses  which 
must  work  intermittently.  They  are  made  in  so  many  forms 
that  a  description  of  them  would  be  beyond  the  scope  of  this 
work.  A  very  full  description  of  many  forms  is  given  in  the 
Transactions  of  the  A.  S.  M.  E.,  Vol.  XXX,  to  which  reference 
has  already  been  made.  Fig.  114  illustrates  the  most  common 
form  of  disengaging  coupling  for  heavy  work.  The  part  B  is 
made  fast  to  the  shaft  to  be  driven,  while  part  A,  which  is  com- 
pelled to  rotate  by  the  feather  F,  can  be  moved  axially  along 
the  driving  shaft.  A  ring  R,  fitting  the  groove  G  loosely  in  a 
radial  direction,  is  connected  by  the  pins  P  to  an  operating  lever 


| 

7 

j- 

/ 

( 

J" 

\ 

1 

: 

... 

__ 

~ 

i 

L  — 

Ni 

i 
^~i    J 

P/ 

s 

S 

114. 


FIG. 


which  is  not  shown.  When  the  part  A  is  moved  forward  till  the 
jaws  J  engage,  A  will  drive  B  positively  in  either  direction.  In 
order  to  facilitate  the  engaging  of  the  jaws  they  are  often  made 
as  in  Fig.  115,  but  in  this  case  the  driving  can  be  in  one  direction 
only.  The  total  cross-sectional  area  of  the  jaws  must  be  such 
that  they  will  not  shear  off  under  the  load,  and  the  area  of  the  jaw 
faces  must  be  sufficient  to  prevent  crushing. 

Frequently,  for  light  work,  only  one  feather  is  used,  but  two 
feathers  are,  in  general,  better,  both  on  account  of  the  driving 
effort  and  for  ease  of  operation. 

129.  Flexible  Couplings.  Where  it  is  desirable  to  have  a 
small  amount  of  flexibility  in  a  shaft,  a  flexible  coupling,  such  as 
is  shown  in  Fig.  116,  is  employed.  These  members  are  much 
used  for  connecting  rapidly  revolving  machines  to  prime  movers, 
as  in  the  case  of  a  dynamo  directly  coupled  to  a  steam  engine, 
the  object  being  to  prevent  undue  stress,  or  bearing  pressure, 
from  lack  of  accurate  alignment  of  the  two  shafts.  In  the  con- 


AXLES,    SHAFTS,    AND    SHAFT    COUPLINGS 


307 


struction  shown,  the  shafts  A  and  B  are  fitted  with  heavy  flanges, 
F,  which  carry  pins,  P.  Links  of  leather  or  other  elastic  material 
connect  pins  on  one  flange  with  pins  on  the  other,  there  being  as 
many  links  as  there  are  pins  in  each  flange.  This  arrangement 


FIG.  116. 

allows  for  a  slight  angle  between  the  axes  of  the  shafts,  or  for  a 
small  lack  of  coincidence  in  the  axes.  The  pins  in  one  disc  are 
sometimes  placed  on  a  smaller  diameter  than  those  on  the  other, 
so  that  in  case  of  failure  of  the  links  the  pins  will  not  strike  and 
cause  breakage. 


CHAPTER  XII 
BELT,  ROPE,  AND  CHAIN   TRANSMISSION 

130.  General  Considerations.  When  power  is  to  be  trans- 
mitted from  one  shaft  to  another,  especially  when  such  shafts 
are  not  far  apart,  in  such  a  manner  that  the  velocity  ratio  of  the 
two  must  be  constant,  some  form  of  toothed  gearing  is  usually 
employed.  When,  however,  it  is  not  necessary  that  the  velocity 
ratio  remain  constant,  flexible  elastic  connectors  are  much  used. 
When  the  distance  through  which  power  is  to  be  transmitted  is 
comparatively  short  (50  feet  or  less),  flat  belts,  or  ropes  of  cotton 
or  manila,  are  most  common;  while  for  longer  distances  steel 
ropes  have  certain  advantages.  For  small  amounts  of  power, 
round  belts  of  leather  are  much  used.  Chain  drives,  which  are 
virtually  flexible  connectors  running  on  toothed  wheels,  have 
lately  come  into  extended  use  for  transmitting  power  over  com- 
paratively short  distances.  They  are  very  efficient,  maintain 
positive  velocity  ratio  between  the  two  shafts,  and  can  be  used 
when  the  distance  between  shafts  is  too  great  for  convenient 
use  of  gears. 

Leather  belts  are  made  by  cementing,  sewing,  or  riveting 
together  strips  of  leather  cut  from  oak-tanned  ox-hides.  Where 
only  one  thickness  is  used  they  are  known  as  single  leather  belts; 
where  two,  three,  or  four  thicknesses  are  needed  to  obtain  a  heavy 
belt,  they  are  known  respectively  as  double,  triple,  and  quadruple 
belts.  Cotton  belts  are  made  either  by  weaving  in  a  loom,  or  are 
built  up  of  several  layers  of  canvas,  sewed  together,  with  a  special 
composition  between  each  fold.  They  are  very  little  used  in 
this  country.  Rubber  belts  are  made  of  several  layers  of  canvas, 
held  together  with,  and  completely  covered  by  a  rubber  com- 
position. They  are  very  effective  in  wet  places.  Belts  of  raw- 
hide are  also  used  to  some  extent. 

308 


BELT,    ROPE,    AND    CHAIN   TRANSMISSION 


309 


The  ends  of  all  belts  are  joined,  to  make  them  continuous, 
either  by  lacing  or  sewing,  or  by  some  kind  of  special  fastening 
of  which  there  are  many  on  the  market,  or  by  making  a  permanent 
joint  by  cementing  and  riveting.  The  latter  method  is  much 
preferable  where  it  can  be  applied,  as  it  makes  the  joint  practically 
as  strong  as  the  rest  of  the  belt,  and  gives  a  smooth  surface  which 
runs  better  than  any  joint.  Other  kinds  of  joints  reduce  the 
strength  of  the  belt  from  60  to  75  per  cent,  but  inasmuch  as  the 
lacing  can  be  replaced  and  the  belt  itself  has  its  life  prolonged  by 
reduced  load,  this  initial  loss  of  efficient  strength  is  not  as  waste- 
ful as  it  at  first  appears. 

131.  Theoretical  Consideration  of  Belts  and  Ropes.  In  Fig. 
117,  let  A  represent  a  pulley  whose  centre  is  at  O,  and  which  is 


ds 


FIG.  117. 

connected  by  a  belt  as  shown  to  the  pulley  B,  whose  centre  is  at 
Ot.  When  no  turning  moment  is  applied  to  the  driving  pulley  A, 
the  tensions  in  the  two  parts  of  the  belt  are  the  same,  except 
possibly  for  friction  of  the  bearings,  and  is  that  due  to  the  initial 
tension  with  which  the  belt  is  placed  upon  the  pulleys.  Let  this 
total  initial  tension  on  each  side  of  the  belt  be  called  T3. 

It  is  evident  that  this  initial  tension  will  cause  the  belt  to 
exert  a  pressure  upon  the  pulley,  and  this  pressure  will  induce 
a  frictional  resistance  opposing  relative  sliding  between  the  belt 
and  the  pulley.  If  now  a  turning  moment  is  applied  to  Ay  and 
a  resisting  moment  to  B,  the  pull  upon  the  belt  due  to  this 
frictional  resistance  will  increase  the  tension  in  the  lower  part 
of  the  belt,  and  decrease  the  tension  in  the  upper  part.  Let 
these  new  total  tensions  be  called  T1  and  T2  respectively.  It  is 
evident  that  the  tendency  of  the  belt  to  slip  around  the  pulley, 


310  MACHINE    DESIGN 

owing  to  the  difference  in  tension  on  the  two  parts  of  the  belt, 
is  resisted  by  the  frictional  resistance  between  the  belt  and  pulley. 
The  difference  in  tensions  tends  to  rotate  the  pulley  B,  and 
when  the  turning  moment  (7\  —  T2)rl  becomes  equal  to  the 
resisting  moment  applied  to  B,  rotation  will  take  place. 

If  the  difference  between  Tl  and  T2  which  is  necessary  to 
overcome  the  resisting  moment,  is  small  compared  to  the 
frictional  resistance  between  the  pulley  and  belt,  no  slipping 
of  the  belt  on  the  pulley  will  occur.  To  obtain  this  result 
in  practice,  would  necessitate  the  use  of  very  large  belts, 
relatively,  for  the  power  transmitted.  It  has  been  found  to 
be  better  practice  to  use  smaller  belts  and  allow  the  belt  to 
slip  somewhat. 

In  addition  to  the  slipping  action  noted  above,  all  belts  are 
subjected  to  what  is  known  as  creep.  Referring  again  to  Fig.  117 
consider  a  piece  of  the  belt  of  unit  length  moving  on  to  the  pulley 
under  a  tension  7\.  As  this  piece  of  belt,  of  unit  length,  moves 
around  with  the  pulley  from  M  to  N,  the  tension  to  which  it  is 
subjected  decreases  from  7\  to  T2  and  the  piece,  owing  to  its 
elasticity,  shrinks  in  length  accordingly.  The  pulley  A,  there- 
fore, continually  receives  a  greater  length  of  belt  than  it  delivers, 
and  the  velocity  of  the  surface  of  the  pulley  is  faster  than  that 
of  the  belt  which  moves  over  it.  In  a  similar  way  the  pulley  B 
receives  a  lesser  length  of  belt  than  it  delivers,  and  its  surface 
velocity  is  slower  than  that  of  the  belt  which  moves  over  its 
surface.  This  creeping  of  the  belt,  as  it  moves  over  the  pulley, 
results  in  some  loss  of  power,  and  is  unavoidable.  The  total  loss 
of  speed  due  to  both  slip  and  creep  should  not  exceed  3%;  that 
is,  the  surface  speed  of  the  driving  pulley  should  not  exceed  that 
of  the  driven  pulley  by  more  than  3%.  Good  practice  limits 
this  value  to  about  2%.  When  the  total  slip  approaches  20%, 
there  is  danger  of  the  belt  sliding  off  of  the  pulley  entirely. 

Since  the  pulling  power  of  a  belt  is  proportional  to  the  differ- 
ence between  Tl  and  T2J  it  is  necessary  to  know  the  relation 
which  exists  between  these  quantities. 

Let  /=  the  tension  per  square  inch  of  belt  section  at  any  point 
on  the  pulley. 


BELT,    ROPE,    AND   CHAIN    TRANSMISSION 


tl  =  the  tension  per  square  inch  of  belt  section  on  the  tight 

side  in  pounds. 
/2  =  the  tension  per  square  inch  of  belt  section  on  the  slack 

side  in  pounds. 
p  =  the  maximum  allowable  tension  per  square  inch  of  belt 

in  pounds. 
/  =  effective  pull  of  belt  per  square  inch  of  cross-section 

"j  (V~~  ^2)  >  in  pounds. 

v  =  the  velocity  of  the  belt  in  feet  per  second. 
w  ==  the  weight  of  one  cubic  inch  of  belt  in  pounds. 
q  =  the     reaction     of     pulley 

against  one  linear  inch  of 

belt    of    the    width    con- 
sidered, in  pounds. 
c  =  the    centrifugal    force     of 

one  cubic  inch  of  belt  in 

pounds  at  the  given  speed. 
/<  =  the  coefficient  of    friction 

between  belt  and  pulley. 
r  =  the    radius   of   the  pulley 

in  inches. 
a  =  the  angle  of   belt    contact 

in  degrees. 

0  •=  the  angle  of  belt  contact  in  radians  =  .0175  a 
The  centrifugal  force   of   one   cubic    inch    of    belt    will    be 


12  WIT 


hence  the  centrifugal  force  of  one   linear   inch  of 

12  wv* 


belt  having  i  square  inch  of  cross  section  will  be 


gr 


Let  the  cross-sectional  area  of  the  belt  be  one  square  inch  and 
consider  an  elemental  portion  of  its  length  as  shown  in  Fig.  118. 
It  is  held  in  equilibrium,  when  slipping  is  impending,  by  the 
following  forces:— 

(a)  The  centrifugal  force  =  c  ds 

(b)  The  radial  reaction  of  the  pulley  against  the  belt  =  q  ds. 

(c)  The  frictional  force  =  /£  q  ds. 

(d)  The  tensions  t  and  /  +  dt. 


312  MACHINE    DESIGN 

Resolving  all  forces  vertically 

do  do 

q  ds  +  c  ds  =  t  sin  —  +  (t  +  d  t)   sin  —  .      .      (i) 

2  2 

Here  d  e  is  so  small  that  sin may  be  taken  as  equal  to 

-  in  radians,  without  appreciable  error,  and  the  product  of  d  t 

do 
and   sin  —  may  be  neglected. 

Hence  (i)  may  be  written 

q  ds  +  c  ds  =  t  de (2) 

12  W1? 

but  c  =  -        -  and  d  s  =  r  d  B 
gr 

12  IV  V*  12  WV2 

.'.  c  ds  = ds  — d  0  =  z  d  e  for  convenience. 

gr  g 

Hence  from  (2)  q  ds  =  t  dd   -z  d  0  =  (t  —  z)  d  6     .      .      .     (3) 
From  equality  of  moments  around  O 

t  +  dt  =  t  +  P-  q  ds 

. ' .  d  t  =  11  q  ds (4) 

Substituting  in  (4)  the  value  of  q  ds  obtained  from  (3) 
d  t  =  {j.  (t  —  z}  d  6 


•"•  /*  IT-^~  =f*fd 

J  L  i  —  2  ^  (? 


°r 


t- 

and  common  log  —   -  =  0.439  ^  6 
— 


0.434  M*  0.0076  M«  k 

=  10  =10  =iofor  convenience     .     (6) 

' 


BELT,   ROPE,   AND    CHAIN    TRANSMISSION  313 

/  =  tv  —  t2     . ' .  t2  =  /!  —  /  and   substituting   this  value  of 
t2  in  (6)  and  reducing 


i,       r      I0>-1 
where  C  = — 

10 


[/1-Z]C      .      .      (8) 

v 

If  (8)  be  multiplied  through  by  -  -  it  will  express  the  horse- 
power (h.p.)  which  a  belt  of  one  square  inch  cross-sectional  area 
will  transmit  or, 

12  =  h.p.  =  ft-z]— "  ...      (9) 

550  550 

132.  Practical  Coefficients.  In  the  above  equations  the 
following  quantities  a,  /*  and  z,  must  be  known  or  assumed  before 
a  solution  for  t^  or  /  can  be  made.  The  angle  of  contact,  a,  can 
be  taken  from  the  drawing  of  the  drive  in  question,  and  some 
allowance  should  be  made  for  the  conditions  of  operation.  Thus 
if  the  belt  is  to  run  in  a  horizontal  position,  with  the  slack  side 
on  top,  the  full  theoretical  value  of  a  may  be  taken.  If,  however, 
the  slack  side  must  be  on  the  bottom  (an  arrangement  which 
should  be  avoided  if  possible)  or  if  the  belt  is  to  be  run  in  a  vertical 
position,  some  reduction  must  often  be  made  in  the  theoretical 
value  of  a  to  allow  for  sagging  of  the  belt.  This  also  applies  to 
belts  running  at  high  speed,  where  centrifugal  force  tends  to 
lessen  the  arc  of  contact. 

The  coefficient  of  friction  //  is  an  exceedingly  variable  quan- 
tity, changing  with  the  character  and  the  condition  of  the  surfaces 
of  contact,  the  initial  tension  of  the  belt,  and  the  rate  of  slip.  It 
has  been  found  by  experiment  that,  within  reasonable  limits,  the 
coefficient  increases  with  the  slip  and  that,  as  before  stated,  a 
maximum  rate  of  slip,  including  creep,  not  in  excess  of  about 
3  per  cent  is  good  practice.  Experiments  made  by  Professor  Dieder- 
richs  in  the  laboratories  of  Sibley  College  gave  the  values  of  /< 


3*4 


MACHINE    DESIGN 


shown  in  the  first  column  of  the  following  table.  Allowing  for  the 
difference  between  conditions  in  the  laboratory  and  those  found 
in  practice,  the  value  shown  in  the  second  column  may  be  used 
in  designing  leather  belts. 

For  pulleys  made  of  pulp,          ^  =  o .  29 .  .  o .  20 

For  pulleys  made  of  wood,         //  =  0.31 o .  22 

For  pulleys  made  of  cast  iron,  ft  =  o .  46 o .  30 

Values  considerably  above  these  were  found  for  paper  pulleys  of 
special  construction. 

The  quantity  z  is  proportional  to  the  weight  of  the  belt  per 
cubic  inch.  For  ordinary  leather  (which  is  most  commonly 
used),  w  may  be  taken  from  0.03  to  0.04,  an  average  value  being 
0.035  pounds. 

Table  XVIII  has  been  calculated  with  a  value  of  w  =  0.035, 
while  Table  XIX  is  abbreviated  from  "Transmission  of  Power 
by  Belting"*  by  Wilfred  Lewis. 

TABLE  XVIII 

12  WV* 


Values  of  z 

V  =  ft.  per  minute,  w 


,  for  v  =  ft.  per  sec.,  or 
035- 


V 

3° 

40 

5° 

60 

70 

80 

90 

100 

no 

120 

130 

140 

V 

z 

i,  800 

2,400 

3,000 

3,600 

4,200 

4,800 

5>400 

6,coo 

6,600 

7,200 

7,800 

8,400 

"•75 

20.9 

32-5 

47.0 

64.2 

83-4 

105-5 

I30-5 

157-6 

187.6 

22O.2 

255-5 

Example.  Design  a  belt  to  operate  a  dynamo  of  15  H.F. 
capacity,  when  the  belt  velocity  is  2,400  ft.  per  minute.  Assume 
fj.  =  0.30,  a  =  180°  and  /x  =  200  Ibs. 

From  equation  (9)  the  horse-power  transmitted  by  a  belt 
having  a  cross-sectional  area  of  one  square  inch  is  for  these  con- 
ditions: 

Cv  61  X  40 

h.p.  =  [tl-z]    ~     =  [200-20.9]  =  7-9 


.*.  the  cross-section  required  =  —  =  i.o  sq.    n. 

7-9 

which  is  equivalent  to  a  belt  -fa"  thick  and  8"  wide. 


*  Transactions  A.  S.  M.  E.,  Vol.  VII,  page  579. 


BELT,    ROPE,    AND    CHAIN   TRANSMISSION 


315 


The  total  tension  (7\)  in  the  tight  side  of  the  belt  will  be 
1.9  X  200  =  380  Ibs.  The  total  tension  (T2)  in  the  slack  side 
will  be  this  value  minus  the  required  effective  pull,  P,  which  is 
found  by  dividing  the  foot  pounds  of  work  to  be  done  by  the 

velocity  of  the  belt  or,  P  =  — 33'°°°  =  206.     Hence  T2  = 


Tl  -  P  =  380  -  206 


2,400 
174  pounds. 


Values  of  C  = 


(Nagle) 


Deg 

rees  of 

Contact 

=  a 

/* 

90 

TOO 

no 

120 

130 

140 

ISO 

1  60 

170 

1  80 

•'5 

.210 

.230 

.250 

.270 

.288 

•3°7 

•325 

•342 

•359 

376 

.20 

.270 

•295 

•319 

•342 

•364 

.386 

.408 

.428 

•  448 

.467 

•25 

•325 

•354 

.381 

.407 

•432 

•457 

.480 

•503 

•524 

•544 

•30 

•376 

.408 

.438 

.467 

•494 

.520 

•544 

•567 

•59° 

.610 

•35 

•423 

•457 

.489 

.520 

•548 

•575 

.600 

.624 

.646 

.667 

.40 

.467 

.502 

•536 

•567 

•597 

.624 

.649 

•673 

695 

•7*5 

•45 

•507 

•544 

•579 

.6lO 

.640 

.667 

.692 

•715 

•737 

•757 

•55 

•578 

.617 

.652 

.684 

•7i3 

•739 

•763 

•785 

.805 

.822 

Equations  (7)  and  (8)  involve  the  relations  which  exist  between 
T1  and  T2  for  a  given  set  of  conditions,  but  they  do  not  indicate 
the  relation  between  them  and  the  initial  tension  Ta.  It  was 
formerly  supposed  that  the  sum  of  7\  and  T2  was  constant  and 
equal  to  2T3;  and  this  relation  may  still  be  used  for  very  rough 
calculations.  Mr.  Wilfred  Lewis*  has  shown,  experimentally, 
that  this  is  not  true.  The  ratio  of  stress  to  strain  in  leather  and 
rubber  increases  with  the  strain  instead  of  being  proportional  to 
it  as  in  ductile  metals.  When  a  belt  transmits  power  the  tension 
is  increased  on  the  tight  side  and  decreased  on  the  slack  side  till 
the  difference  in  tension  is  equal  to  the  required  driving  force. 


*  See  Transactions  A.  S.  M.  E.,  Vol.  VII,  page  566. 


31 6  MACHINE    DESIGN 

This  is  accomplished  by  what  virtually  amounts  to  shortening 
the  belt  on  the  tight  side,  a  given  amount,  by  transferring  this 
amount  to  the  slack  side.  Because,  however,  of  the  relation 
between  stress  and  strain  noted  above,  the  increase  of  tension  on 
the  tight  side,  due  to  this  amount  of  shortening,  is  greater  than 
the  decrease  of  tension  on  the  slack  side  due  to  an  equal  amount 
of  lengthening,  and,  as  a  consequence,  the  sum  of  the  two  ten- 
sions is  increased*  as  the  effective  pull  is  increased.  Sugges- 
tion: Place  a  rubber  band  over  the  fingers  of  the  two  hands 
and  stretch  it  moderately;  then  twist  one  of  the  hands  in  either 
direction  and  the  increase  of  force  tending  to  bring  the  hands 
together  will  be  apparent. 

In  the  case  of  a  long  horizontal  belt  the  increase  in  the  sum 
of  the  tensions  is  still  further  augmented  in  driving,  because 
the  tension  on  the  slack  side  (with  a  proper  initial  tension  in  the 
belt)  is  largely  due  to  the  sag  of  the  belt  from  its  own  weight; 
and  thus  the  tension  on  the  slack  side  tends  to  remain  nearly 
constant,  while  the  tension  on  the  tight  side  increases  with  the 
power  transmitted,  at  a  given  speed.  It  is  found  that  the  sum 
of  the  tensions  on  the  two  sides,  when  driving,  may  exceed  the  sum 
of  the  initial  tensions  by  about  33  percent  in  vertical  belts,  and  in 
horizontal  belts  the  increase  may  be  limited  only  by  the  strength 
of  the  belt.  In  addition  to  the  causes  discussed,  the  tension  on 
both  parts  of  the  belt  are  increased  by  the  centrifugal  action  due 
to  the  mass  of  that  portion  of  the  belt  which  is  rotating  round 
the  pulley  axis.  This  latter  cause  increases  the  stresses  on  both 
the  tight  and  slack  sides  of  the  belt,  and  decreases  adhesion  be- 
tween the  belt  and  the  pulley,  but  does  not  increase  the  loads  on 
the  shafts  which  produce  pressure  at  the  bearings  and  flexure  of 
the  shafts. 

Large  belts  should  therefore  be  put  on  with  care,  as  to  initial 
tension.  Ordinarily,  the  initial  tension  is  left  to  trained  judg- 
ment, but  it  would  seem  that  the  more  advanced  practice  of 
splicing  the  belt  under  a  known  initial  tension  will  add  to  the 
life  of  large  and  important  belts. 

*  See  Transactions  A.  S.  M.  E.,  Vol.  VII,  page  569. 


BELT,    ROPE,    AND    CHAIN   TRANSMISSION  317 

133.  Strength  of  Belting.     The   ultimate   strength   of  good 
leather  belting  will  vary  from  3,500  to  6,000  pounds  per  square 
inch.     Professor  Benjamin  *  gives  the  strength  of  cotton  belting  as 
about  the  same  as  good  leather.     He  also  found  that  four-ply 
rubber  belting  had  a  tensile  strength  of  from  840  to  930  pounds 
per  inch  of  width.     The  ultimate  strength  of  belting  seldom 
enters  as  a  factor  in  belt  design,  as  the  real  strength  of  the  belt 
is  in  the  joint.     Where  the  ends  of  the  belt  are  laced  together,  a 
maximum  working  stress  of  200  to  300  pounds  per  sq.  inch  is 
found  to  be  good  practice;    and  where  the  belt  is  cemented  to- 
gether, thus  making  it  "  endless,"  a  working  stress  of  400  pounds 
per  square  inch  may  be  used.     The  thickness  of  leather  belting 
varies  from  T3g-  to  -^  inch  for  single  leather,  and  from  $4  to  % 
inch  for  double  leather.     Hence  for  single  leather, 

p  =  50  to  75  pounds  per  inch  of  width  for  laced  belts. 

p  =  100  pounds  per  inch  of  width  for  cemented  belts. 
For  double  leather  belts  p  may  be  taken  at  twice  these  values. 
Lower  stresses  than  these  are  often  advocated,  and  undoubtedly 
lower  stresses  increase  the  life  of  the  belt. 

134.  Velocity  of  Belting.     In  equation  (8)  when  z  =  tl}  f  =  c 
and  the  belt  will  exert  no  turning  force,  the  centrifugal  force 
relieving  all  frictional  resistance  between  the  belt  and  pulley. 

If  tl  be  taken  as  high  as  400  pounds,  and  w  =  .035  this  will 

12  Wl? 

occur  when  z  =  400  or  when  -         -  =  400  whence  v  =  175  ft. 

o 

per  second  or  10,500  feet  per  minute. 

If  equation  (8)  be  multiplied  through  by  v,  the  velocity  of 
the  belt,  it  will  express  the  rate  at  which  energy  is  being  de- 
livered, or 

r~        12  w  v2~i 
fV-V[tl-z]    C=V[V-     -— ]C 

If  now  //  =  .3,  w  =  .035,  a  =  180,  which  are  average  conditions, 
the  equation  becomes 

j  v  =  v  [/!  -  .013  v2]  X  0.6  =  0.6  /!  v  —  .0078  v3 

*  See  "Machine  Design,"  by  Benjamin,  page  186. 


318  MACHINE    DESIGN 

Differentiating  the  right-hand  side  with  respect  to  v  and  equating 
to  zero 

0.6 /x  —  .0234^  =  o  or  v  =  5.1  V^  •  •  (10) 
which  gives  the  relation  between  v  and  tl  for  maximum  power. 
When  ^  =  400,  v  =  102  feet  per  second  or  6,120  feet  per  minute 
and  when  tl  =  275  pounds,  v  =  85  feet  per  second,  or  5,100  feet 
per  minute.  It  is  often  necessary  to  run  belts  at  much  lower 
speeds  than  these;  but  it  is  not  economical  to  exceed  these  limits. 
A  speed  of  a  mile  per  minute  may  be  taken  as  about  the  economi- 
cal maximum  limit;  and  it  so  happens  that  this  is  also  about  the 
limit  of  safety  for  ordinary  cast-iron  pulley  rims.  For  durability 
combined  with  efficiency,  a  speed  of  3,000  to  4,000  feet  per 
minute  may  be  taken  as  a  fair  value,  though  practical  limitations 
such  as  speed  of  shafting  and  diameter  of  pulleys  often  fix  belt 
velocities  at  much  lower  values. 

135.  Efficiency  of  Belting.     The  losses  of  power  in  belt  trans- 
mission consist  of  the  loss  due  to  slip  and  creep,  that  due  to  bend- 
ing the  belt  over  the  pulley,  and  the  frictional  losses  at  the  shaft 
bearings,  due  to  belt  pull.     The  first  two,  slip  and  creep,  should 
not  exceed  3  per  cent,  and  2  per  cent  is  better.     The  loss  due  to 
bending  the  belt  is,  usually,  negligible  although  the  effect  on  the 
life  of  thick  belting  running  on  small  pulleys  is  important.     The 
losses  at  the  bearings  may  be  considerable  if  the  belt  must  be 
laced  on  under  great  initial  tension  in  order  to  carry  the  load,  and 
this  condition  should  be  avoided  except  where  it  is  absolutely  nec- 
essary to  use  a  short   belt.     A  well-designed  belt  transmission 
should  have  an  efficiency  at  least  as  high  as  95. per  cent,  and  it 
may  be  as  high  as  97  per  cent  including  bearing  losses. 

136.  Other  Equations,  Common  Rules.     If  in  equation   (9), 
w  be  taken  as  0.032  and  tl  as  305  pounds  the  equation  reduces  to 

h-P*  =  [  -55  —  0.0000216  v2]vC     .     .     .     (n) 


iok-i 


where  C  =  -  — —  as  before  and  h.  p  =  horse-power  per  square 

inch  of  belt  area.  If  the  equation  be  multiplied  by  A,  the  area  of 
the  belt  cross-section,  it  will  express  the  total  horse-power  trans- 
mitted, or  H.  P.  =[  .55  -  0.0000216  v2]  v  CA  ....  (12) 


BELT,    ROPE,    AND    CHAIN    TRANSMISSION  319 

Professor  Diederichs  has  pointed  out  that  equation  (12)  is  iden-r 
tical  with  that  reported  by  Mr.  Nagle  to  the  A.  S.  M.  E.*  and 
commonly  known  by  his  name.  Values  of  C  have  already  been 
given  in  Table  XIX. 

In  the  transactions  of  the  American  Society  of  Mechanical 
Engineers,  January,  1909,  Mr.  Carl  Earth  presents  a  more  ex- 
tended mathematical  treatment  of  the  driving  capacity  of  belts. 
He  also  presents  scientific  methods  for  measuring  the  tension 
in  belting.  Many  other  formulae  of  a  strictly  empirical  char- 
acter are  given  by  different  authorities  and  some  of  them 
are  very  convenient.  In  general  these  last  formulae  neglect 
centrifugal  action  and  are  hence  applicable  only  to  belt  speeds 
below  2,500  feet  per  minute.  Thus  a  common  rule  is  that  a 
single  leather  belt  one  inch  wide  traveling  1,000  feet  per 
minute  will  transmit  i  H.P.  Kent's  "  Mechanical  Engineer's 
Pocket  Book,"  page  877,  gives  a  number  of  these  so-called 
practical  rules. 

137.  Practical  Considerations.  One  of  the  most  valuable 
contributions  to  the  literature  of  the  subject  is  "  Notes  on 
Belting,"  by  Mr.  F.  M.  Taylor,  in  Vol.  XV  of  the  Transac- 
tions of  the  American  Society  of  Mechanical  Engineers.  Mr. 
Taylor  kept  an  accurate  record  of  measurements  and  observa- 
tions on  belts  in  use  at  the  Mid  vale  Steel  Co.'s  works,  for  nine 
years,  and  gives  many  valuable  facts  and  practical  suggestions 
in  his  paper.  A  satisfactory  abstract  of  it  is  not  possible  here. 
Mr.  Taylor  advocates  thick  narrow  belts  rather  than  thin  wide 
belts,  t  He  sums  up  his  investigation  in  36  "Conclusions," 
among  which  are : 

"A  double  leather  belt  having  an  arc  of  180°  will  give  an 
effective  pull  on  the  face  of  the  pulley  per  inch  of  width  of  belt 
of  35  pounds  for  oak-tanned  and  fulled  leather,  or  30  pounds  for 
other  types  of  leather  belts  and  6-  to  7 -ply  rubber  belts." 

"The  number  of  lineal  feet  of  double  belting,  i  inch  wide, 

*  Vol.  II,  page  91. 

f  While  in  general  this  conclusion  is  justifiable,  care  should  be  taken  that  it 
is  not  carried  to  the  extreme  where  the  life  of  the  belt  may  be  shortened  by  ex- 
cessive bending. 


320  MACHINE    DESIGN 

passing  around  a  pulley  per  minute,  required  to  transmit  one 
horse-power  is  950  feet  for  oak-tanned  and  fulled  leather  belt, 
and  1,100  feet  for  other  types  of  leather  belts,  and  6-  to  y-ply 
rubber  belts." 

"The  most  economical  average  total  load  for  double  belting, 
is  65  to  73  pounds  per  inch  of  width,  i.e.,  200  to  225  pounds  per 
square  inch  of  section.  This  corresponds  to  an  effective  pulling 
power  of  30  pounds  per  inch  of  width." 

"The  speed  at  which  belting  runs  has  comparatively  little 
effect  on  its  life,  till  it  passes  2,500  or  3,000  feet  per  minute." 

"The  belt  speed  for  maximum  economy  should  be  from  4,000 
to  4,500  feet  per  minute." 

It  should  be  especially  noted  that  Mr.  Taylor  advocates  a 
maximum  belt  tension  of  about  one-half  that  ordinarily  used. 
This  would,  of  course,  increase  the  first  cost  of  the  installation 
materially.  His  values,  however,  are  not  based  on  the  minimum 
size  of  belt  required  to  simply  transmit  a  given  horse-power,  but 
on  the  size  of  belt  which  will  transmit  that  horse-power  for  a  given 
time  with  minimum  wear  and  loss  of  time  due  to  breakage  or 
taking  up  to  restore  tension.  Whether  his  practice  is  followed 
or  not,  it  indicates  the  true  aspect  of  the  problem,  and  is  a  step 
in  advance. 

In  laying  out  belt  drives,  care  should  be  taken  to  keep  the 
diameters  of  pulleys  reasonably  large.  The  constant  bending 
action  to  which  the  belt  is  subjected  as  it  runs  around  the  pulley 
is  a  great  source  of  wear,  and  where  the  pulley  is  very  small, 
compared  to  the  thickness  of  the  belt,  this  may  be  excessive. 
For  this  reason  also  it  is  probably  better  to  run  the  hair  side  of 
the  belt  next  to  the  face  of  the  pulley  as  this  side  is  more  easily 
cracked  by  bending,  than  the  flesh  side,  which  is  more  soft 
and  pliable.  Mr.  Taylor  says  it  is  safe  to  run  double  leather 
belts  on  pulleys  12  inches  in  diameter. 

The  total  length  of  the  belt  or  distance  between  shaft  centres 
also  deserves  attention.  A  belt  being  elastic,  acts  like  a  spring 
when  tension  is  applied  to  it.  The  longer  the  belt  the  greater 
will  be  the  total  stretch  for  a  given  load.  Suddenly  applied  loads, 
therefore,  produce  less  stress  in  long  belts  than  in  short  ones 


BELT,    ROPE,    AND    CHAIN   TRANSMISSION  321 

(see  Art.  24).  If,  however,  the  distance  between  centres  is  too 
great,  compared  to  the  size  of  the  belt,  the  belt  is  liable  to  flap 
and  run  unevenly  on  the  pulleys.  For  small,  narrow  belts  a 
maximum  distance  of  15  feet  is  good  practice,  while  for  heavier 
belts  25  feet  is  found  satisfactory. 

A  number  of  important  investigations  of  belt  transmission 
have  been  reported  to  the  American  Society  of  Mechanical 
Engineers.  See  the  following  papers  in  the  transactions  of  the 
Society  by:  Mr.  A.  F.  Nagle,  Vol.  II,  page  91;  Professor  G. 
Lanza,  Vol.  VII,  page  347;  Mr.  Wilfred  Lewis,  Vol.  VII,  page 
549;  Mr.  F.  W.  Taylor,  Vol.  XV,  page  204;  Professor  W.  S. 
Aldrich,  Vol.  XX,  page  136.  Abstracts  of  these  as  well  as  other 
valuable  data  are  given  in  Kent's  "  Mechanical  Engineer's  Pocket 
Book,"  pages  876  to  887. 

FIBROUS  ROPE  DRIVES 

138.  General  Considerations.  When  the  amount  of  power 
to  be  transmitted  is  large,  the  width  of  belt  required  may  be 
excessive,  even  when  the  belt  is  made  very  thick.  To  run  wide 
belts  successfully,  the  shafting  must  be  kept  in  perfect  parallel 
alignment,  and  the  distance  between  shaft  centres  must  not  be 
too  great.  For  these  reasons  rope  drives  have  been  found  very 
satisfactory  where  the  amount  of  power  to  be  transmitted  is 
large,  and  the  distance  of  transmission  relatively  great.  They 
are  also  particularly  serviceable  for  connecting  shafts  which  are 
not  parallel,  as  in  the  case  of  "  quarter-turn "  drives,  especially 
where  a  belt  would  have  to  be  of  considerable  width  and  would, 
as  a  consequence,  run  badly. 

In  all  fibrous  rope  drives  the  surf  aces  of  the  pulleys  or  "sheaves" 
are  provided  with  wedge-shaped  grooves  to  receive  the  rope  and 
thereby  give  the  rope  a  better  grip  on  the  sheave.  For  drives 
of  moderate  length,  40  to  150  feet,  fibrous  ropes  of  cotton,  hemp 
or  manila  fibre  are  chiefly  employed.  For  transmitting  power 
comparatively  great  distances,  wire  rope  is  more  common,  al- 
though fibrous  ropes  are  also  used  for  comparatively  long  trans- 
missions. In  all  long-distance  transmission  the  rope  must  be 
supported  at  intervals  by  idler  pulleys. 

21 


322 


MACHINE    DESIGN 


Fig.  119*  shows  a  typical  rope  drive  where  the  line  shafting 
of  each  floor  of  a  mill  is  driven  by  its  own  rope  drive  from  the 
main  shaft  of  the  engine. 

139.  Materials  for  Fibrous  Ropes.  Round  ropes  of  leather, 
or  rawhide,  are  used  to  a  limited  extent,  when  the  amount  of 
power  to  be  transmitted  is  small.  Rawhide  is  especially  useful 
in  damp  places,  but  since  it  costs  about  six  times  as  much  as 
vegetable  fibre  rope,  its  application  is  very  limited.  Leather 
belts  or  ropes  of  square  f  or  wedge-shaped  section  have  also  been 


FIG.  119. 

used  to  a  limited  extent.  In  certain  localities  in  Great  Britain, 
hemp,  which  is  a  local  product,  is  quite  extensively  used;  but 
cotton  and  manila  fibre  are  by  far  the  most  common  for  trans- 
missions of  any  considerable  size.  In  this  country  manila  fibre  is 
used  almost  exclusively,  while  in  England  and  on  the  Continent 
cotton  rope  is  also  much  employed. 

It  is  obvious  that  as  a  twisted  rope  of  any  fibrous  material 
bends  while  passing  over  the  sheave,  there  must  be  a  certain 

*  Reproduced  by  permission  from  "  The  Blue  Book  of  Rope  Transmission." 
f  For  a  fuller  discussion  of  such  ropes  see  "  Machine  Design,"  by  H.  J.  Spooner. 


BELT,    ROPE,    AND   CHAIN    TRANSMISSION  323 

amount  of  internal  friction.  The  result  of  this  action  is  very 
noticeable  in  any  old  manila  rope  which  has  been  used  without 
lubrication.  When  such  a  rope  is  broken  open  it  is  found  to  be 
filled  with  powdered  fibre,  due  to  the  internal  chafing.  For  this 
reason  manila  fibre,  which  is  naturally  rough,  is  usually  lubri- 
cated, while  being  twisted  into  rope,  -with  tallow,  paraffine, 
soapstone,  graphite,  or  some  such  lubricant. 

Cotton  fibres,  on  the  other  hand,  are  smoother  and  hence  give 
rise  to  less  internal  friction.  They  are,  therefore,  usually  laid 
up  dry  into  rope,  a  dressing  or  lubricant  being  applied  to  the 
exterior  to  prevent  small  fibres  from  rising  on  the  outside,  thus 
starting  the  rope  to  fraying.  This  dressing  also  excludes  mois- 
ture and  retains  the  natural  oils  in  the  interior  fibres.  Cotton 
fibre  is  not  as  strong  as  manila. 

Professor  Flather*  makes  the  following  comparison  between 
cotton  and  manila  rope:  "  As  compared  with  manila,  then,  the  ad- 
vantages of  cotton  ropes  of  the  same  diameter  are:  Greater 
flexibility,  greater  elasticity,  less  internal  wear  and  loss  of  power 
due  to  bending  of  the  fibres,  and  the  use  of  smaller  pulleys  for  a 
given  diameter  of  rope.  Its  disadvantages  are:  Greater  first 
cost,  lesser  strength,  and  possibly  a  greater  loss  of  power  due  to 
pulling  the  ungreased  rope  out  of  the  groove — in  any  case  this 
is  usually  small  with  speeds  over  2,000  feet  per  minute." 

140.  Theoretical  Considerations.  The  general  equations 
(7),  (8),  and  (9),  of  Art.  131,  which  were  deduced  for  flat  belts 
hold  also  for  round  ropes  if  the  proper  notation  be  substituted. 
In  these  equations  the  unit  mass  of  belt  was  taken  as  one  cubic 
inch.  With  ropes  it  is  more  convenient  to  take  a  piece  of  rope 
one  inch  in  length  and  one  inch  in  diameter.  With  the  following 
exceptions,  therefore,  the  notation  used  here  will  be  the  same  as 
that  used  in  Art.  131. 

Let  it/  =  the  weight  of  a  piece  of  rope  i  inch  in  diameter  and 
i  inch  long. 

12  Wf  V2 

Let  z'  =  -           -  where  w'  has  the  value  above. 
g 

*  "  Rope  Driving,"  by  J.  J.  Flather,  page  81. 


324 


MACHINE    DESIGN 


Let  t\  =  the  tension  in  a  rope  of   i  inch  diameter  on  the 
tight  side. 

Let  Cf  =  a  new  coefficient  =  C  modified  on    account    of 
wedging  effect  of  groove. 
Then  equations  (8)  and  (9)  become 

'J  =  [l\-z']C'      .     .  .     (13) 

andh.p.  -[/',  -  *' 


In  equations  (8)  and  (9)  the  frictional  force  between  the 
pulley  and  the  belt  for  a  flat  belt  is  taken  as  jj.  q  where  q  is  the 
radial  pressure  between  the  pulley  and  the  belt.  In  a  grooved 
pulley  the  pressure  between  the  pulley  and  the  rope  is  greater 

e 

than  the  radial  pressure  in  the  ratio  of  cosec  -  to  unity,  where 

0  is  the  angle  between  the  sides  of  the  groove.     The  frictional 

0 
resistance  between  the  rope  and  sheave  is  therefore  /JL  q  cosec  —  . 

0 
If  p.  cosec  ~  be  substituted  for  fj.  in  the  quantity  C  (equations 

8  and  9)  the  result  Cr  may  be  used  as  indicated  in  equations 
(13)  and  (14)  for  rope  drives.  The  value  of  /z  for  rope  sheaves 
has  not  been  determined  with  any  degree  of  accuracy.  Professor 
Flather*  after  reviewing  what  experimental  data  there  is  on  the 
subject,  concludes  that  0.12  is  a  fair  value  and  computes  the 

6  B 

following  values  of  $  =  p  cosec  —  =  o.  1  2  cosec  — 

TABLE  XX 

e 

0  =  coefficient  of  friction  =  0.12  cosec  — 


Angle  of  groove. 

30° 

35° 

40° 

45° 

50° 

55° 

60° 

* 

.46 

.40 

•35 

•3i 

.28 

.26 

•  24 

It  is  obvious  that  if  (j>  be  used  instead  of  p  in  Table  XIX,  the 
corresponding  values  of  C  in  Table  XIX  will  be  the  new  constant 

*  "  Rope  Driving,  '  page  112. 


BELT,    ROPE,    AND    CHAIN   TRANSMISSION  325 

C.  Thus  if  e  =  45°,  <£  =  .31.  If  also  a  =  180°,  C'  from  Table 
XIX  =  .61  about.  The  angle  45°  has  been  found  to  be  the  most 
satisfactory  and  is  most  commonly  used.  If  the  angle  0  be  less 
than  45°,  the  wedging  action,  hence  the  pulling  capacity  is  in- 
creased, but  the  power  loss  and  wear  of  rope  due  to  drawing 
it  out  of  the  grooves  is  greater.  For  such  sheaves,  with  6  =  45° 
and  a  =  180° 

h,p  =  .6i[t\-z']^-o  .      (15) 

As  before  stated,  reliable  data  on  the  coefficient  of  friction 
for  ropes  are  scarce,  and  designing  engineers  have  approached 
the  problem  of  rope  drives  without  regard  to  this  coefficient. 
One  of  the  most  important  contributions  to  the  subject  is  that  of 
Mr.  C.  W.  Hunt  (see  Transactions  A.  S.  M.  E.,  Vol.  XII).  The 
notation  of  Mr.  Hunt's  article  has  been  changed  somewhat  to 
correspond  with  that  used  in  this  text. 

Let  d  =  diameter  of  the  rope  in  inches. 
d  =  sag  of  rope  in  inches. 
L  =  distance  between  pulleys  in  feet. 
•u/  =  weight  of  one  inch  of  rope  of  one-inch  diameter. 
W  =  weight  of  one  foot  of  rope  of  diameter  d. 
Tl  =  total  tension  in  rope  on  tight  side. 
T2  =  total  tension  in  rope  on  slack  side. 
T0  =  tension  necessary  to  give  the  rope  adhesion. 
K  =  the  total  tension  applied  to  each  side  of  the  rope  due 

to  centrifugal  force. 
P  =  effective  turning  force  =  7\  —  T2 

Then  7\  =  T0  +  K  +  P 

and  T2  =  T0  +  K 

Mr.  Hunt  says  that  "when  a  rope  runs  in  a  groove  whose 
sides  are  inclined  toward  each  other  at  an  angle  of  45°  there  is 
sufficient  adhesion  when  Tl  -f-  T2  =  2.  However,  he  assumes  a 
somewhat  different  ratio  in  the  development  of  his  equation,  for 
which  he  assumes  "that  the  tension  on  the  slack  side  necessary 
for  giving  adhesion  is  equal  to  one-half  the  force  doing  useful 
work  on  the  driving  side  of  the  rope." 


326  MACHINE    DESIGN 

Or  T  =  -  and  7\  =  T  +K  +  P  =  -  +  K  +  P  =  ^P  +  K 

2  22 

P 

and  T2  =  To  +  #  =  -  +  K  by  assumption. 


If  equation  (16)  be  multiplied  through  by  --  it  will  express  the  total 
horse-power  transmitted  or 


The  tension  K  on  each  side  of  the  rope  for  an  arc  of  contact  of 

12  Wf  V2 

1  80°  and  a  rope  of  one  inch  diameter  is  --  ,  which  is  iden- 

o 

tical  with  the  constant  z'  in  equation  (14).     Mr.  Hunt's  formula 
therefore  may  be  written 


where  h.p.  is  the  horse-power  transmitted  by  a  rope  one  inch  in 
diameter.  This  is  identical  in  form  with  the  theoretical  equation 
(15)  and  differs  from  it  only  by  a  negligible  amount  in  the  value 
of  the  coefficient. 

It  would  seem  therefore  that  Mr.  Hunt's  assumptions  give 
results  very  close  to  those  obtained  by  using  the  value  0.12  for  ^ 
as  recommended  by  Professor  Flather. 

It  is  to  be  noted  that  the  values  of  z  given  in  Table  XVIII 
may  be  used  in  computing  values  of  z'.  The  quantities  are  the 
same  except  for  the  weight  w'  '.  In  Table  XVIII,  w  =  the  weight 
of  one  cubic  inch  of  leather  =  .035.  Inequation  (18),  w'  =  the 
weight  oj  one  inch  of  rope  oj  one  inch  diameter  =  .028  for  manila 
rope  and  .022  for  cotton  rope.  If,  therefore,  the  values  given  in 

Table  XVIII  are  multiplied  by  -  they  are  applicable  to  manila 
ropes,  and  if  multiplied  by  —  they  may  be  used  for  cotton  ropes. 


BELT,    ROPE,    AND   CHAIN   TRANSMISSION  327 

Example.  What  diameter  of  manila  rope  is  necessary  to 
transmit  25  H.P.  when  running  4,000  feet  per  minute,  in  grooves 
having  an  angle  of  45°.  Take  t\  =  200  pounds,  and  w'  =  .028. 
From  Table  XVIII,  z,  for  the  given  velocity  =  64  nearly.  .  *  .  z1  = 

64  x  —  =  51.     From  equation  (18)  the  horse-power  which  a  rope 

0 

one  inch  in  diameter  will  deliver  under  these  conditions  is 


-  P.   =  [<',    -  *]  -  [200   - 


25 
.  '  .  the  cross-section  required  =  -  =  twice  the  area  of  a  one- 

12.  1 

inch  rope  which  corresponds  to  a  rope  i-Hs"  in  diameter. 

Fig.  120*  shows  curves  based  on  equation  (17),  giving  the  total 
horse-power  transmitted  by  ropes  of  various  sizes  for  T1  =  2ood2, 
and  will  be  found  convenient  for  making  calculations. 

141.  Strength  of  Fibrous  Ropes.     The  ultimate  strength  of 
manila  transmission  ropes  may  be  taken  as  about  7,oood2  and 
for  cotton  rope  as  about  4,6ood2  where  d  =  diameter  of  rope  in 
inches.     The  working  stress  must  be  taken  very  much  less  than 
these  values  or  otherwise  the  life  of  the  rope  is  much  shortened. 
For  manila  rope  Mr.  Hunt  recommends  that  the  working  tension 
(7\)  be  not  over  200  d2.     The  same  factor  of  safety  would  give 
130  d2  as  the  allowable  working  tension  for  cotton  ropes;    but 
since  cotton  ropes  are  somewhat  less  affected  by  internal  chafing 
the  working  tension  may,  perhaps,  be  safely  taken  at  a  rather 
higher  value. 

142.  Velocity    of    Fibrous    Ropes.     The    centrifugal    force 

1  2  Wr  V2 

produces  a  tension  in  a  rope  of  one  inch  diameter  of  zr  =  ~ 

or  in  a  rope  of  diameter  d  the  centrifugal  force  =  -  —  .     The 

Q 

allowable  stress  in  the  rope  is  200  d2.     The  centrifugal  force  will 

12  w'  (P  1? 

equal  the  allowable  tensile  stress  when  -  =  200  d2  or 

g 

*  From  "The  Blue  Book  of  Rope  Transmission,"  by  the  American  Mfg.  Co. 


328 


MACHINE  DESIGN 


when  v  =  140  feet  per  second,  at  which  speed  the  effective  pull 
becomes  zero  for  this  allowable  working  stress. 

If  equation  (18)  be  differentiated  and  the  differential  be  equated 
to  zero  as  in  Art.  134,  the  resultant  equation  will  give  the  value 


FIG.  120. 


of  the  velocity  where  the  work  done  is  a  maximum,  for  a  rope 
of  one  inch  in  diameter.  This  is  found  to  be  about  4,900  feet 
per  minute.  Since  the  centrifugal  force,  and  the  total  working 
stress,  both  vary  as  the  area  of  the  rope  this  limiting  velocity 


BELT,    ROPE,    AND   CHAIN   TRANSMISSION 


329 


applies  to  all  sizes  of  ropes,  a  conclusion  which  is  borne  out  by 
the  curves  of  Fig.  120. 

It  has  been  found,  in  practice,  that  the  most  economical  speed 
for  ropes  is  from  4,000  to  5,000  feet  per  minute.  If  speeds 
greater  than  this  are  used,  the  wear  on  the  rope  is  excessive. 
For  a  fixed  value  of  Tl  =  200  d?  the  first  cost  of  a  rope  is  a  mini- 
mum at  about  4,900  feet  as  above,  and  this  first  cost  is  greater 
by  10  per  cent  if  the  velocity  is  increased  to  6,000,  or  decreased  to 
3,700  feet  per  minute.  The  first  cost  is  increased  50  per  cent 
when  the  velocity  is  reduced  to  2,400  feet  per  minute  with 


FIG.  121. 

7\  =  200  d2  but  the  reduction  in  speed  increases  the  life  of  the 
rope. 

143.  Systems  of  Rope-Driving.  There  are  two  methods  of 
placing  fibrous  ropes  on  the  sheaves.  In  the  Multiple,  or  English 
system,  several  separate  ropes  run  side  by  side,  each  rope  forming 
a  closed  circuit  in  exactly  the  same  manner  as  a  flat  belt,  and 
running  constantly  in  its  own  particular  groove  on  each  pulley. 
In  the  Continuous  or  American  system  one  rope  only  is  used,  the 
rope  being  carried  continuously  from  one  pulley  to  the  other  till 
all  the  grooves  are  filled,  and  it  is  then  spliced ;  so  that  the  rope 
as  it  leaves  the  last  groove  of  the  driven  sheave  is  returned 
to  the  first  groove  of  the  driver,  or  driving  pulley,  by  means 
of  an  idler,  or  guiding  sheave.  This  idler  is  usually  arranged 
so  that  through  it  a  suitable  tension  may  be  put  upon  the  rope 
(see  Fig.  121). 


330  MACHINE   DESIGN 

Regarding  the  merits  of  the  two  systems  it  may  be  said  that 
the  multiple  system  is  the  simpler,  and  that  it  also  provides 
considerable  security  against  the  loss  of  time  due  to  breakdowns, 
as  it  is  not  likely  that  more  than  one  rope  will  break  at  a  time. 
When  failure  of  a  rope  does  occur,  the  broken  rope  may  be 
removed  and  repaired  at  a  more  convenient  opportunity,  allowing 
the  other  ropes  to  carry  the  load  temporarily.  Occasionally, 
however,  the  breaking  of  a  rope  in  the  multiple  system  may  cause 
great  delay,  on  account  of  the  broken  rope  becoming  entangled 
in  one  of  the  rope  sheaves  and  winding  up  upon  it  before  the 
machinery  can  be  stopped.  In  this  system  the  individual  ropes 
must  be  respliced  occasionally  to  take  up  the  sag  in  the  rope  due 
to  stretching.  The  velocity  ratio  transmitted  by  a  new  rope  will 
be  different  from  that  transmitted  by  an  old  one  which  has  worn 
smaller,  and  hence  fits  d^wn  farther  into  the  grooves,  thereby 
changing  its  effective  radius.  The  velocity  ratio  of  the  two 
sheaves  can,  however,  have  but  one  value,  and,  therefore,  the 
tendency  will  be  for  either  the  old  or  the  new  ropes  to  carry  the 
whoie  load.  When  the  driving  sheave  is  the  larger,  this  will 
result  in  a  tendency  to  throw  more  load  on  the  old  ropes;  when 
the  driving  sheave  is  the  smaller  the  tendency  is  to  throw  more 
load  on  the  larger  and  new  ropes.  The  unequal  speed  of  the 
ropes,  of  course,  leads  to  unequal  stress;  and  slipping  and  con- 
sequent wear  are  sure  to  occur. 

The  continuous  system  is  more  flexible  in  its  application  than 
the  multiple  system;  for,  owing  to  the  limited  sag  in  the  ropes 
due  to  the  action  of  the  weighted  idler,  the  rope  may  be  run 
safely  at  any  angle.  This  form  of  drive  is,  therefore,  much 
used  for  vertical  and  quarter-turn  drives,  and,  generally,  where 
the  transmission  is  of  a  complicated  nature.  The  principal 
objections  to  the  system  are  the  danger  of  loss  of  time  due  to  a 
breakdown,  and  the  unequal  straining  of  the  various  spans  of 
the  rope  particularly  with  a  varying  load  or  inequality  of  grooves. 
When  a  load  is  suddenly  applied  to  the  continuous  system  all 
the  spans  on  the  slack  side  become  slacker  except  that  which 
runs  over  the  idler  and  which  is  kept  at  a  fixed  tension.  A  much 
greater  load  is  hence  brought  on  the  driving  span  of  rope  next  to 


BELT,   ROPE,    AND   CHAIN  TRANSMISSION 


331 


the  idler  and  some  time  must  elapse  before  this  load  can  be 
equalized  over  all  the  spans.  Mr.  T.  Spencer  Miller*  has  pointed 
out  that  the  general  tendency  to  unequal  straining  may  be  some- 
what obviated,  where  the  sheaves  are  of  different  diameters,  by 
making  the  angle  of  the  groove  in  the  small  sheave  somewhat 
sharper  than  that  in  the  larger,  so  that  the  product  of  the  arc  of 
contact  and  the  cosecant  of  half  the  groove  angle  are  equal; 
thus  making  the  tendency  to  slip  equal. 

The  above  are  the  principal  points  of  difference  between  the 
two  systems.  The  particular  conditions  of  the  installation  must 
be  considered  in  making  a  choice  between  them. 

144.  Sheaves  for  Fibrous  Ropes.  The  sheaves  over  which 
ropes  are  to  run  deserve  special  attention.  Care  should  be  taken 


FIG.  122  (a). 


FIG.  122  (b). 


FIG.  122  (c). 


that  the  form  of  the  grooves,  and,  the  effective  diameters  are  the 
same  for  all  grooves  of  the  same  sheave  and  the  surfaces  should 
be  accurately  finished  and  well  polished,  as  any  roughness  or 
unevenness  seriously  affects  the  life  of  the  rope.  As  the  result 
of  much  experimentation  two  forms  of  grooves  as  shown  in  Fig. 
122  (a)  and  122  (b)  have  become  most  common.  In  Fig.  122  (b) 
the  sides  of  the  groove  are  straight  while  in  122  (a)  the  sides  are 
curved.  This  curving  of  the  sides  makes  the  angle  of  the  groove 
somewhat  flatter  at  the  bottom  and  hence  when  the  rope  has  been 
reduced  in  diameter  from  wear  it  lies  lower  in  the  groove  and  will 
slip  a  little  more  readily  than  when  it  is  new  anjd  occupies  a  higher 
position.  This  is  of  importance  in  relieving  the  old  rope  of  a 
tendency  to  pull  harder  as  indicated  in  the  preceding  article. 
The  curved  outline  is  also  said  to  assist  the  rope  to  roll  in  the 

*  Transactions  A.  S.  M.  E.     Vol.  XII,  page  243. 


332  MACHINE    DESIGN 

groove,  a  very  desirable  feature  since  it  distributes  the  wear  on 
the  rope.  The  curved  groove  is  therefore  much  used  in  the 
multiple  system.  In  the  continuous  system  the  rope  necessarily 
rotates  as  it  passes  round  the  idler  to  the  first  groove. 

The  angle  of  the  groove,  as  before  stated,  is  usually  45°. 
The  grooves  of  idler  pulleys  for  simply  supporting  the  rope  when 
the  stretch  is  great  are  not  made  v-shaped  but  as  shown  in  Fig. 

122   (c). 

The  wear  of  fibrous  ropes  is  both  internal  and  external,  the 
internal  wear  being  due  largely  to  chafing  of  the  fibres  on 
each  other  in  bending  the  rope  over  the  sheaves.  For  this  rea- 
son sheaves  should  be  as  large  as  possible,  and,  in  general, 
should  not  have  a  diameter  less  than  forty  diameters  of  the  rope. 

145.  Deflection  or  Sag.  Where  the  span  between  the  pulleys 
is  considerable  the  amount  of  deflection  is  sometimes  of  import- 
ance. Since  the  deflection  varies  with  the  distance  between 
pulleys,  the  size  and  speed  of  the  rope  and  the  difference  in 
elevation  of  the  pulleys,  it  is  impossible  to  express  the  relation 
existing  between  them  in  a  single  formula.  For  the  simple  case 
of  the  horizontal  drive  the  approximate  deflection  on  the  driving 
side  may  be  determined  both  for  the  continuous  and  multiple 
systems  and  also  the  deflection  of  the  slack  side  of  the  continuous 
system,  where  uniform  tension  is  maintained  by  a  tension  weight. 
In  the  multiple  system,  however,  ample  allowance  must  be  made 
on  the  slack  side,  as  new  ropes  stretch  very  rapidly,  and  the 
deflection  may  become  excessive  before  resplicing  can  be  per- 
formed. Mr.  Hunt  gives  the  following  equation  (transformed), 
for  computing  the  deflection  in  horizontal  drives: 


A    =  _-  -  x    -L ±_    .        .        .        .        (in) 

2W       >  4  JF2        8 

Where  T  is  the  total  tension  on  either  the  slack  or  tight  side  de- 
pending on  the  side  for  which  it  is  desired  to  compute  the  deflec- 
tion, W  the  weight  of  rope  per  foot,  L  the  span  in  feet  and  A  the 
deflection  in  feet.  Where  the  tension  on  the  driving  side  is 
assumed  to  be  equal  to  200  d2,  regardless  of  speed,  the  deflection 
on  the  driving  side  will  be  constant  for  a  given  span.  As  the 


BELT,    ROPE,    AND    CHAIN    TRANSMISSION 


333 


tension  in  the  rope  due  to  centrifugal  action  increases  as  the  square 
of  the  velocity,  there  is  an  increasing  total  tension  T2  on  the  slack 
side  for  a  fixed  value  of  7\;  and  hence  the  deflection  on  the  slack 
side  decreases  with  the  velocity,  the  span  remaining  constant. 
The  value  of  T2  may  be  computed  and  substituted  in  equation 
(19)  to  find  the  deflection. 

Mr.    Frederick    Green  *    gives    the  following    approximate 
formula  for  computing  the  deflection : 


A    = 


W  X  L2 

ST 


(20) 


Where  the  symbols  are  the  same  as  in  equation  (19),  and  from 
which  he  has  calculated  the  following  table  on  the  assumption 
that  7\  =  200  d\ 


TABLE  XXI 


SAG  ON  SLACK  SIDE. 

Distance 
between 
Pulleys, 

Sag  on 
Driving  Side, 
All  Speeds, 

Velocity,  Feet  per  Minute. 

Feet. 

Feet. 

3,000 

4,000 

4,500 

S.ooo 

5,500 

3° 

.19 

•45 

•39 

•36 

•33 

•3° 

40 

•34 

.80 

.69 

.64 

•59 

•53 

50 

•53 

I  .2 

i  .1 

I  .0 

•92 

.84 

60 

.76 

1.8 

i-7 

1.4 

!-3 

I  .2 

7° 

I  .0 

2.4 

2  .1 

1.9 

.     i-7 

1.6 

80 

1.4 

3-2 

2.9 

2-5 

2-3 

2  .  I 

90 

i-7 

4.0 

3-5 

3-2 

3-° 

2-7 

100 

2  .1 

5-° 

4-3 

4.0 

3-7 

3-3 

120 

3-° 

7.2 

6.2 

5-7 

5-3 

4.8 

140 

4-i 

9.9 

8-5 

7-8 

7.2 

6.6 

1  60 

5-4 

12.9 

ii  .  i 

10.2 

9-5 

8.6 

WIRE-ROPE  TRANSMISSION 

146.  General.  Ropes  made  of  iron  or  steel  wire  have  been 
used  to  a  considerable  extent  for  transmitting  power  over  com- 
paratively great  distances.  The  introduction  of  electrical  trans- 
mission has,  however,  greatly  curtailed  the  field  as  far  as  power 
transmission  is  concerned;  although  wire  ropes  are  still  much 


*  See  "The  Blue  Book  of  Rope  Transmission,"  by  American  Mfg.  Co. 


334  MACHINE   DESIGN 

used  for  conveying  materials  such  as  coal,  rock,  etc.,  by  means 
of  buckets  attached  at  intervals  along  the  rope.  The  rope  in 
such  installations  moves  at  very  low  velocities  and  constitutes  a 
different  problem  from  that  of  power  transmission.  Wire  ropes 
are  also  much  used  for  hoisting  work  such  as  elevator  and  mine 
work  and  for  carrying  static  loads  as  in  supporting  smokestacks, 
masts  and  suspension  bridges. 

147.  Materials  for  Wire  Ropes.     Wire  ropes  are  usually  made 
of  wrought  iron,  open  hearth  steel,  or  crucible  steel.     For  very 
severe  work  especially  strong  crucible  steel  known  as  plough  steel 
is  used.     For  a  few  special  cases,  copper  and  bronze  are  employed. 

The  John  A.  Roebling's  Sons  Co.  publications,  give  the  fol- 
lowing values  for  the  tensile  strength  of  various  kinds  of  wire. 

Swedish  Iron    45,ooo  to  100,000  Ibs.  per  sq.  in. 

Open  Hearth  Steel    50,000  to  130,000  Ibs.  per  sq.  in. 

Crucible  Steel 130,000  to  190,000  Ibs.  per  sq.  in. 

Plough  Steel     190,000  to  350,000  Ibs.  per  sq.  in. 

They  also  state  that  it  is  difficult  to  obtain  from  a  sample  of  rope  in 
a  testing  machine,  more  than  90  per  cent  of  the  aggregate  strength 
of  all  the  wires.  This  is  due  to  the  difficulty  of  getting  a  perfect 
grip  on  the  rope  so  that  all  the  wires  will  carry  their  full  share  of 
the  load ;  and  also  because  the  inner  wires  of  a  strand  are  shorter 
than  the  outer  wires  and  are  therefore  more  quickly  overloaded. 
The  wires,  on  account  of  the  twisted  construction,  also  tend  to 
mutually  cut  into  each  other,  thus  rendering  them  more  liable 
to  fracture  under  heavy  loads.  On  account  of  this  latter  action 
ropes  made  with  a  short  twist  break  at  a  lower  percentage  of 
their  full  strength  than  those  of  a  longer  twist. 

148.  Power  Transmission  by  Wire  Rope.     Wire  ropes   for 
power  transmission  are  usually  made  of  iron  or  soft  steel  and  are 
laid  up  with  a  soft  core  of  hemp  in  order  to  give  greater  flexibility. 
They  cannot  be  run  on  metallic  surfaces  and  the  sheaves  must  be 
lined  at  the  bottom  with  soft  rubber  or  similar  yielding  material. 
Great  care  must  be  taken  that  the  rope  does  not  chafe  and,  un- 
like the  sheaves  for  fibrous  ropes,  the  grooves  in  sheaves  used 
for  wire  rope  are  so  formed  that  the  sides  of  the  groove  do  not 
compress  the  ropes.     In  wire-rope  sheaves,  the  radius  at  the 


BELT,    ROPE,    AND   CHAIN    TRANSMISSION  335 

bottom  of  the  groove  is  always  greater  than  that  of  the  rope 
itself  so  that  wire-ropes  drive,  like  flat  belts,  simply  through 
the  friction  on  the  bottom  of  the  groove,  due  to  the  tension  of 
the  rope.  The  lining  of  the  bottom  of  the  groove  (leather,  wood, 
or  some  other  comparatively  soft  material)  gives  increased  friction 
as  well  as  less  wear  of  the  rope.  The  sheaves  should  be  as  large 
as  possible  to  minimize  the  bending  effect  on  the  rope;  one 
hundred  rope  diameters  being  often  taken  as  the  minimum 
diameter  of  the  sheave. 

The  general  theory  and  equations  developed  for  fibrous  rope 
hold  also  for  wire  rope,  proper  constants  being  substituted.  It 
is  evident  from  this  discussion  that  wire  ropes  can  safely  trans- 
mit a  greater  amount  of  power  than  fibrous  ropes  of  the  same 
diameter,  because  of  the  much  higher  allowable  tensile  stress. 

The  table  on  the  following  page,  which  is  taken  from  a  circular 
of  the  John  A.  Roebling's  Sons  Co.,  shows  the  power  that  may 
be  transmitted  by  iron  ropes  of  various  sizes  with  sheaves  of 
different  diameters  and  rotative  speeds.  These  values  are  for  a 
rope  made  with  six  strands  around  a  hemp  core,  each  strand 
consisting  of  seven  wires.  This  table  does  not  make  allowances 
for  the  change  of  stress  due  to  the  change  of  centrifugal  force  at 
various  speeds;  but  it  does  consider  the  influence  of  the  sheave 
diameter  on  the  bending  stress.  For  example :  a  W  rope  on  an 
eight-foot  sheave  running  100  r.p.m.,  transmits  only  32  H.P.; 
while  the  same  rope  transmits  64  H.P.,  when  running  on  a  ten- 
foot  sheave  at  80  r.p.m.  or  at  the  same  linear  velocity.  By  re- 
ferring to  Fig.  120  it  is  seen  that  a  manila  rope  of  \%" 
diameter  transmits  only  30  H.P.,  at  the  most  economical  velocity, 
or  at  about  twice  the  velocity  in  the  above  instance. 

For  hoisting  and  for  transmission,  if  the  sheave  diameters 
must  be  much  smaller  than  those  given  in  the  preceding  table,  a 
more  flexible  rope  is  used.  This  consists  of  six  strands  around  a 
hemp  core,  but  each  strand  is  made  up  of  19  wires,  which  are,  of 
course,  of  smaller  diameter  than  those  used  for  corresponding  sizes 
of  seven-wire  strands.  The  lining  of  the  bottoms  of  the  grooves 
in  the  sheaves  should  be  maintained  in  good  repair.  If  it  be- 
comes irregular,  through  wear,  the  rope  may  be  bent  at  a  sharp 


336 


MACHINE   DESIGN 


TABLE  XXII. 

TABLE  OF  TRANSMISSION  OF  POWER  BY  WIRE  ROPES  * 


Diameter  of 
Wheel  in  Feet. 

Number  of 
Revolutions. 

1 

|i 
1* 

H 

Diameter  of 
Rope. 

y 

PH 
K 

Diameter  of 
Wheel  in  Feet. 

Number  of 
Revolutions. 

n  . 

3  (U 

Diameter 
of  Rope. 

Horse-Power. 

3 

80 

23 

f 

3 

7 

140 

2O 

T9, 

35 

3 

IOO 

23 

1 

3t 

8 

80 

19 

1 

26 

3 

120 

23 

f 

4 

8 

IOO 

19 

1 

32 

3 

I4O 

23 

f 

4i 

8 

120 

19 

1 

39 

4 

80 

23 

1 

4 

8 

140 

19 

i 

45 

4 

IOO 

23 

i 

5 

9 

80 

(  20 
1  19 

|   T96   f 

J   47 
(   48 

4 

120 

23 

1 

6 

9 

IOO 

[*> 

(  19 

}"•*  1 

i    58 
|    60 

4 

140 

23 

f 

7 

9 

120 

(20 
1  19 

[  T9c  i 

j.   69 
1    73 

5 

80 

22 

iV 

9 

9 

140 

|  T96   I 

(   82 
i    84 

5 

IOO 

22 

lV 

ii 

10 

80 

JI9 

1  18 

[f   ii 

j    64 
f    68 

(  10 

K   11 

j    80 

5 

120 

22 

aV 

J3 

10 

IOO 

1  18 

\  1  H 

f   85 

5  ' 

140 

22 

A 

15 

10 

120 

i  18 

h« 

3   96 

/    102 

6 

80 

21 

1 

14 

10 

140 

(  ig 
1  18 

[  1   iJ- 

i  "I 

6 

IOO 

21 

1 

17 

12 

80 

\* 

[  H  I 

\    93 
1    99 

6 

120 

21 

i 

20 

12 

IOO 

(11 

\  ii  t 

(   116 

(  17 

(   124 

6 

140 

21 

i 

23 

12 

120 

1" 

}  ii  f 

j   140 
1   149 

7 

80 

2O 

T9<T 

2O 

12 

120 

16 

1 

173 

j  8 

)     , 

j   141 

7 

IOO 

20 

A 

25 

14 

80 

1  7 

j"  I  J* 

i  148 

7 

120 

20 

30 

14 

IOO 

n 

j  i  4 

i  176 

<  185 

*  Taken  from  a  publication  of  the  John  A.  Roebling's  Sons  Company,  of 
Trenton,  N.  J. 

The  above  table  gives  the  power  transmitted  by  Patent  Rubber-lined  Wheels 
and  Wire  Belt  Ropes,  at  various  speeds. 

Horse -powers  given  in  this  table  are  calculated  with  a  liberal  margin  for  any 
temporary  increase  of  work. 


BELT,    ROPE,    AND   CHAIN    TRANSMISSION  337 

angle  in  passing  over  the  high  spots  of  the  lining,  with  a  resultant 
increase  in  the  stress  of  the  wires.  This  last  action,  however,  is 
not  equivalent,  so  far  as  the  life  of  the  rope  is  concerned,  to  run- 
ning over  a  correspondingly  smaller  sheave,  for  every  portion  of 
each  wire  is  bent  around  each  sheave  once  during  every  circuit 
of  the  rope;  while  it  is  not  likely  that  the  same  portion  of  the 
rope  will  frequently  come  in  contact  with  any  single  irregularity 
in  the  lining. 

ROPES  AND  CABLES  FOR  HOISTING 

149.  Fibrous  Ropes  for  Hoisting.  In  power  transmission  it  is 
usually  possible  to  install  sheaves  large  enough  to  prevent  the 
bending  action  from  seriously  affecting  the  life  of  the  rope;  but 
in  hoisting  work  this  is  not  always  possible,  on  account  of  the 
size  and  clumsiness  of  the  resulting  tackle.  Thus,  a  manila 
rope  of  i  inch  diameter,  if  used  for  power  transmission,  should 
run  over  a  sheave  at  least  40  inches  in  diameter  but  if  used  for 
hoisting  it  might  be  required  to  run  over  a  block  sheave  1 2  inches 
or  even  8  inches  in  diameter.  The  internal  friction  and  external 
chafing  are,  in  such  cases,  very  great  and  the  life  of  the  rope, 
even  when  working  at  a  lower  stress,  is  greatly  shortened;  but 
in  hoisting  tackle,  the  frequency  with  which  any  portion  of  the 
rope  passes  over  the  sheaves  is  much  less  than  is  ordinarily  the 
case  in  power  transmission,  on  account  of  lower  speed. 

Theoretical  considerations  are  of  little  or  no  help  in  hoisting 
installations,  and  recourse  must  be  had  to  successful  practice  on 
which,  fortunately,  there  are  considerable  data.  The  following 
table,  from  a  paper  presented  by  Mr.  C.  W.  Hunt,  before  the 
A.  S.  M.  E.,  gives  the  results  of  a  long  series  of  observations,  and 
indicates  the  most  economical  size  of  rope  for  a  given  load.  It 
has  been  found,  by  experience,  that  ropes  larger  or  smaller  than 
those  recommended  in  the  table  are  shorter-lived  under  the  load 
indicated.  The  speeds  indicated  in  the  table  are  defined  as 
follows : 

"Slow" — Derrick,  crane,  and  quarry  work;  50  to  100  feet 
per  minute. 

22 


338 


MACHINE    DESIGN 


"Medium" — Wharf  and  cargo   work;    150   to  300  feet  per 
minute. 

"Rapid" — 400  to  800  feet  per  minute. 


TABLE  XXIII 

WORKING    LOAD   FOR  MANILA    ROPE 


A. 

B. 

C. 

D. 

E. 

F. 

G. 

H. 

Diame- 

Ulti- 

WORKING LOAD  IN  POUNDS. 

MINIMUM  DIAMETER  OF  SHEAVES 
IN  INCHES. 

ter  of 

mate 

Rope, 
Inches. 

Strength, 
Pounds. 

Rapid. 

Medium. 

Slow. 

Rapid. 

Medium. 

Slow. 

I 

7,100 

200 

400 

1,000 

40 

12 

8 

9,000 

250 

500 

1,250 

45 

J3 

9 

11,000 

300 

600 

1,500 

5° 

14 

10 

13,400 

380 

75° 

1,900 

55 

15 

ii 

15,800 

45° 

900 

2,200 

60 

16 

12 

18,800 

53° 

1,100 

2,600 

65 

i? 

13 

21,800 

620 

1,250 

3,000 

70 

18 

14 

150.  Wire  Hoisting  Ropes.     On  overhead  travelling  cranes, 
elevators  and  mine  work,  iron  or  steel  cables  are  used  almost 
exclusively,  as  here  it  is  usually  possible  to  install  sheaves  or 
drums  of  large  diameter.     For  rough  service,  deep  mine  work  or 
wherever  great  strength  is  necessary,  these  ropes  are  sometimes 
made  of  crucible  or  plough  steel.     Great  care  should  be  exercised 
in  installing  such  ropes  and  it  is  well,  in  general,  to  obtain  the 
advice  of  the  manufacturers  before  selecting  any  rope  made  of 
crucible  steel,  especially  if  great  safety  is  desired.     A  factor  of 
safety  of  at  least  5  should  be  used  in  ordinary  work,  and  for 
elevator,  or  similar  work,  a  factor  as  high  as  10  or  15  is  some- 
times desirable.      Table  XXIV,  taken  from  a  publication  of  the 
John  A.  Roebling's  Sons  Co.,  gives  data  on  standard  hoisting 
ropes.     For  open-hearth  steel  the  strength  as  given  for  iron  rope 
may  be  increased  25  per  cent.     It   will  be  noticed   that   these 
tables  are  based  on  a  factor  of  safety  of  5. 

CHAINS  AND  CHAIN  TRANSMISSION 

151.  Chains  may  be  conveniently  divided  into  three  classes: 

(a)  Chains  for  raising  and  supporting  loads. 

(b)  Chains  for  conveying  purposes. 

(c)  Chains  for  power-transmission  purposes. 


BELT,    ROPE,    AND   CHAIN   TRANSMISSION 


339 


Chains  for  Hoists.  In  the  first  class  are  such  chains  as  are 
used  on  cranes  and  hoisting  appliances.  Chains  of  this  character 
are  made  with  elliptical-shaped  links  and  should  be  manufactured 
of  the  best  wrought  iron  to  insure  perfect  welding  where  the  link 
is  joined.  The  links  themselves  should  be  as  small  as  possible 
to  minimize  the  collapsing  action  or  bending  due  to  the  pull  of 
the  adjacent  links,  and  also  that  due  to  winding  the  chain  upon  a 
circular  drum.  Such  chains  are  sometimes  called  short-link, 
close,  or  crane  chains. 

TABLE  XXIV 

STRENGTH    OF    IRON    AND    STEEL    HOISTING    ROPES 


SWEDISH  IRON. 

CRUCIBLE  STEEL. 

c 

%£ 

& 

^Od 

d 

fe  • 

J 

^"0  £ 

Diameter  i 
Inches. 

Weight  p 
Foot  in  11 

l-sts 

60.g 

Minimum 
Diameter 
Sheaves  i 
Feet. 

Diameter 
Inches. 

ii 

II 

||| 

Minimum 
Diameter  i 
Sheaves  ii 
Feet. 

2f 

n-95 

114 

22.8 

16 

2f 

n-95 

228 

45  -6 

10 

2I 

95 

18.9 

15 

2i 

9-85 

190 

37-9 

9i 

2f 

8.00 

78 

15  .60 

13 

2i 

8.00 

156 

31.2 

8* 

2 

6.30 

62 

12  .40 

12 

2 

6.30 

124 

24.8 

8 

4-85 

48 

9  .60 

10 

1 

4-85 

96 

19.2 

7i 

4.15 

42 

8.40 

8* 

1 

4-i5 

84 

16.8 

6J 

3-55 

36 

7.20 

7* 

^ 

3-55 

72 

14.4 

5l 

3.00 

6  .20 

7 

1 

3.00 

62 

12.4 

2-45 

25 

5  -°° 

6J 

1 

2-45 

5° 

IO.O 

5 

2  .00 

21 

4  .20 

6 

J 

2  .00 

42 

8-4 

4i 

I.58 

17 

3-40 

Si 

1-58 

34 

6.8 

4 

| 

I  .20 

2  .60 

4! 

1 

I  .20 

26 

S-2 

3! 

| 

0.89 

9-7 

i-94 

4 

t 

0.89 

19.4 

3-88 

3 

|^ 

o  .62 

6.8 

1.36 

3i 

1 

O.62 

13.6 

2.72 

2i 

T\ 

0.50 

5-5 

I  .  10 

2f 

o  .50 

II  .0 

2  .20 

if 

1 

°-39 

4.4 

0.88 

1 

°-39 

8.8 

I.76 

TV 

0.30 

3-4 

0.68 

2 

A 

0.30 

6.8 

I.36 

if 

1 

0  .22 

'  2-5 

0.50 

Ji 

i 

0  .22 

5  -° 

I  .00 

i 

1 

0.15 
0.  10 

I  .2 

0.34 
o  .24 

i 
t 

xi 

0.15 
0.10 

3-4 
2.4 

0.68 
0.48 

t 

The  strength  of  a  chain  link  in  tension  is  less  than  twice  the 
strength  of  a  bar  of  the  iron  from  which  the  chain  is  made  on 
account  of  the  bending  action  due  to  the  manner  in  which  the 
load  is  applied,  and  also  on  account  of  the  weld.  If  W  =  the 
breaking  load  in  pounds,  and  d  =  the  diameter  in  inches  of  the 


340  MACHINE  DESIGN 

bar  from  which  the  link  is  made,  then  the  following  empirical 
equation  may  be  used  for  iron  crane  chains. 

W  ==  54,000  d*     ......     \      .....     (i) 

The  working  load  (W)  should  not  be  more  than  one-third  this 
value  or  W  =  18,000  d2  .  .  ....  .  -  .  .  -  .  (2) 

In  many  cases  a  lower  stress  than  indicated  by  (2)  should  be 
adopted.  Whenever  the  load  is  not  a  direct  pull,  but  severe 
bending  stresses  are  also  induced,  as  in  chain  "slings"  for  handling 
heavy  iron  castings,  the  chain  should  have  great  excess  of  strength. 
Chains  should  be  carefully  inspected  and  tested  or  "proved" 
before  using.  The  "  proof  "  usually  applied  is  one-half  the  ulti- 
mate load.  Where  chains  are  used  for  hoisting  work,  they  are 
likely  to  become  badly  strained.  Annealing  by  heating  allows 
a  readjustment  of  the  structure  of  the  iron,  and  this  should  be 
done  periodically  with  all  such  chains,  particularly  chains  used 
for  slings.  This  also  affords  an  opportunity  to  thoroughly  inspect 
chains  which  are  greased  in  operation.  The  uncertainty  regard- 
ing the  exact  condition  of  a  chain  in  service,  and  the  fact  that  it 
gives  no  warning  of  weakness,  but  may  break  at  a  load  below  the 
normal  working  load,  have  caused  them  to  be  largely  replaced,  on 
such  appliances  as  overhead  cranes,  by  steel  rope.  The  state 
of  the  strength  of  the  latter  is  more  easily  determined  by  inspec- 
tion. 

Weldless*  steel  chain  rolled  from  a  bar  of  special  shape 
has  lately  come  into  use  to  some  extent.  The  chain  is  made  in 
lengths  of  from  60  to  90  feet,  and  the  lengths  are  joined  together 
by  a  link  made  of  special  welding  steel.  They  are  said  to  be 
much  stronger  than  iron  chains. 

152.  Chain  Drums  and  Sheaves.  Drums  on  which  crane 
chains  are  to  wind  should  be  carefully  grooved  so  that  alternate 
links  lie  flat  on  the  surface  of  the  drum;  and  should  have  sufficient 
capacity  to  receive  the  chain  in  one  layer,  as  overwinding  brings 
severe  stresses  on  the  parts  wound  upon  the  drum.  The  diam- 
eter of  the  drum  should  in  no  case  be  less  than  twenty  times  the 
diameter  of  the  chain  used,  and  thirty  times  this  diameter  is  better. 

*  See  "Machine  Design."  by  H.  J.  Spooner,  page  452. 


BELT,   ROPE,    AND   CHAIN   TRANSMISSION 


341 


Where  it  is  not  possible  to  have  the  chain  wind  upon  a  drum, 
pocket  chain  wheels  are  often  used.  These  wheels  are  made  with 
pockets  around  the  periphery  into  which  the  links  fit.  The 
links  are  prevented  from  coming  out  by  a  guide  over  a  portion  of 
the  wheel;  and  hence  cannot  slip  on  the  sheave.  Anchor  chains, 
and  the  chains  of  certain  forms  of  chain  blocks  for  raising  weights, 
run  over  such  sheaves. 

153.  Hoisting-Hooks.     The   hooks   used    for    raising   heavy 
weights  deserve  special  attention.     They  are  usually  made  of 


FIG.  123  (a). 


FIG.  123  (b). 


steel  or  iron  forgings  although  steel  castings  are  employed  to 
some  extent.  If  the  stress  in  the  hook  can  be  kept  low  the  use  of 
steel  castings  may  be  justified;  but  where  the  load  is  great  and 
the  fibre  stress  in  the  hook  necessarily  high,  to  avoid  clumsy 
proportions,  the  hook  should  be  forged  from  ductile  material. 
Let  the  hook  in  Fig.  123  (a)  be  subjected  to  a  vertical  load  P; 

then  XY,  the  most  dangerous  section,  is  apparently  acted  upon 

p 
by  a  direct  stress  p'  =  —  (where  A  is  the  area  of  the  section)  and 

A. 

by  a  flexural  stress  p"  due  to  the  moment  Pa;  the  stress  f  being 


342  MACHINE   DESIGN 

tensile  at  X  and  compressive  at  F.  The  theory  of  article  19, 
therefore  applies,  apparently,  and  equation  M  (Table  VI)  may 
be  used  to  design  the  section,  or 

P       Pae 


Experience  shows  that  members  of  this  kind,  even  when 
made  of  materials  whose  tensile  and  compressive  elastic  limits 
are  about  the  same,  almost  invariably  yield  to  rupture  on  the 
tension  side;  and  the  section  is  usually  made  as  shown,  the 
gravity  axis  being  located  nearer  the  load,  thus  decreasing  the 
tensile  stress  and  increasing  the  compressive  stress,  as  computed 
by  the  above  equation.  Recent  investigations*  have  shown  that 
in  a  curved  beam  loaded  in  this  manner,  the  neutral  axis  does 
not  coincide  with  the  gravity  axis,  as  in  straight  beams,  but  is 
located  nearer  the  tension  side,  and  the  above  theory  is  therefore 
defective,  as  the  true  tensile  stress  is  greater  than  that  given  by 
equation  M.  That  this  is  true  is  borne  out  by  the  fact  men- 
tioned above,  that  hooks  fail  in  tension  when  designed  with  an 
apparent  compressive  stress  considerably  above  the  tensile  stress, 
although  the  elastic  limit  for  either  stress  is  about  the  same. 
The  application!  of  the  more  accurate  theory  is,  however,  some- 
what complicated  and  it  is  believed  that  equation  M  may  be 
safely  used  if  due  care  is  taken  in  assigning  the  limits  of  stress. 

Hooks  for  small  cranes  and  hoists  are  much  more  likely  to  be 
loaded  frequently  to  their  full  capacity  than  hooks  for  raising 
large  loads;  thus  a  hook  on  a  five-ton  crane  may  be  loaded  to  its 
full  capacity  several  times  every  day,  while  the  hook  of  a  twenty-ton 
crane  would  be  thus  loaded  at  rare  intervals.  The  stresses  in 
small  hooks  must  therefore  be  kept  low,  and  fortunately  this  can 
be  done  without  making  the  hook  clumsy.  As  the  size  of  the 
hook  increases,  however,  the  stresses  must  necessarily  be  in- 
creased to  avoid  clumsiness,  but  the  larger  the  hook  the  less 
frequently  will  it  be  fully  loaded  and  a  working  stress  as  high  as 

*  See  "Strength  of  Materials,"  by  Slocum  and  Hancock. 

t  See  "  On  a  Theory  of  the  Stresses  in  Crane  and  Coupling  Hooks.  With  Ex- 
perimental Comparisons  with  Existing  Theory,"  by  Professor  Karl  Pearson  and 
Mr.  E.  S.  Andrews.  Messrs.  Dulan  &  Co. 


BELT,    ROPE,    AND   CHAIN   TRANSMISSION  343 

15,000  pounds  per  square  inch,  or  more,  is  as  safe  in  a  fifty-ton 
hook  as  10,000  pounds  per  square  inch  would  be  in  a  ten-ton 
hook.  (See  Art.  25.) 

The  most  valuable  data  on  crane  hooks  is  that  given  by  Mr. 
Henry  R.  Towne  in  his  "Treatise  on  Cranes,"  as  a  result  of  both 
mathematical  and  experimental  work.  Fig.  123  (a)  and  the 
following  formulas  give  the  most  important  dimensions  of  a  hook 
according  to  this  work,  and  these  proportions  have  been  much 
used  with  uniform  success.  The  basis  for  each  size  is  a  com- 
mercial size  of  round  iron  or  dimension  A.  In  the  following 
formula  A  is  the  nominal  capacity  of  the  hook  in  tons  of  2,000 
pounds.  The  dimension  D  is  assumed  arbitrarily  but  so  as 
to  provide  ample  room  for  the  slings.  The  following  measure- 
ments are  then  expressed  in  inches: 

D  =  .5  A  +  1.25         H  =  i.oSA        K  =  1.13  A 
G  =  .^D  I  --I.M  A         L  =  i.osA 

The  following  gives  the  capacity  of  the  hooks  made  from 
various  sizes  of  bar  stock : 

TABLE  XXV 


Capacity  of  hook  in  tons  

i 

i 

\ 

i 

i* 

2 

3 

4 

5 

6 

8 
~* 

10 

Size  of  bar  A  in  inches  

t 

H 

I 

iiV 

il 

If 

if 

2 

2* 

2| 

3i 

It  is  to  be  noticed  that  the  stresses  allowed  by  Mr.  Towne's 
proportions  are  very  low.  Thus  in  a  ten-ton  hook  the  dimension 
A  is  3  %"  or,  allowing  for  finishing,  the  dimension  B  may  be  taken 
as  3",  which  would  give  a  tensile  stress  in  the  shank  of  only  3,000 
pounds  per  square  inch.  It  should  be  borne  in  mind,  however, 
that  hooks  are  subjected  to  much  abuse  and  the  designer  has  no 
assurance  that  they  will  always  be  loaded  with  a  true  axial  load, 
for  improper  arrangement  of  the  sling  often  throws  the  load  more 
toward  the  point  of  the  hook  and  the  member  is  called  upon  to 
carry  a  bending  moment  greatly  in  excess  of  that  for  which  it  is 
intended. 

When,  however,  hooks  larger  tlun  those  covered  by  Mr. 
Towne' s  work  are  to  be  designed  his  proportions  lead  to  clumsy 
dimensions.  Thus  a  twenty-ton  hook  would  require  a  shank 


344 


MACHINE   DESIGN 


"  in  diameter  and  a  fifty-ton  shank  would  be  6%"  in  diameter. 
Fig.  123  (b)  shows  a  twenty-ton  hook  of  Norway  iron  which  has 
been  successfully  used  in  practice.  The  threaded  shank  being 
3>6'/  in  diameter  is  therefore  stressed  to  about  6,000  pounds  per 
square  inch  but  yet  is  only  as  large  as  the  shank  of  a  ten-ton  hook 
as  given  by  Mr.  Towne's  dimensions.  Examination  of  current 
practice  and  measurements  taken  from  a  number  of  large  hooks 
in  successful  service  indicate  an  allowable  tensile  stress  at  X, 
as  computed  by  equation  M  of  Table  VI,  ranging  from  10,000 
Ibs.  per  square  inch  in  ten-ton  hooks,  to  15,000  Ibs.  per  square 
inch  in  fifty-ton  hooks. 

154.  Conveyor  Chains.  Chains  for  conveying  and  elevating 
materials,  such  as  grain,  coal,  ashes,  etc.,  are  usually  made  of 


FIG.  124  (a). 


FIG.  124  (b). 


FIG.  124  (c). 


malleable  iron,  the  links  hooking  together  in  some  manner.  This 
style  of  chain  is  known  as  link  belt.  On  account  of  the  diverse 
purposes  to  which  they  are  applied,  they  are  made  in  many  forms, 
and  the  selection  of  the  particular  form  for  a  given  problem  is 
usually  made  in  conference  with  the  manufacturer  or  taken  from 
trade  catalogues  giving  the  desired  information.  This  form  of 
chain  is  also  extensively  used  in  rough  machinery,  such  as  agri- 
cultural implements,  for  the  transmission  of  power.  In  such 
cases  the  chains  must  be  run  at  low  speeds,  as  they  become  noisy 
and  unreliable  even  at  moderate  velocities. 

155.  Chains  for  Power  Transmission.  The  chains  heretofore 
discussed  move,  necessarily,  at  low  velocities,  but  of  late  a  demand 
has  arisen  for  chains  which  may  be  run  at  high  speeds  for  the 
purpose  of  transmitting  power.  Such  chains  are  used  when  a 


BELT,    ROPE,    AND    CHAIN    TRANSMISSION 


345 


positive  velocity  ratio  must  be  maintained  between  the  connected 
shafts,  and  where  the  distance  between  shafts  is  so  great  as  to 
make  tooth  gearing  inconvenient.  Of  this  class  there  are  at 
present  three  principal  types,  namely,  roller  chains,  block  chains, 
and  so-called  silent  chains.  Fig.  124  (a)  illustrates  the  simplest 
form  of  roller  chain  in  which  the  pin  A  is  riveted  fast  in  the  outer 
links,  and  rotates  in  the  inner  links.  The  roller  R  lessens  the 
friction  against  the  tooth.  In  this  form  of  chain  the  wear  between 
the  pin  and  the  inner  link  is  excessive,  and  for  this  reason  it  is 
now  little  used  for  power  transmission.  It  is  sometimes  made 
without  the  roller  and  with  several  inside  links  and  is  then  known 
as  stud  chain.  In  this  form  it  is  used  for  very  low  velocities  only. 


FIG.  125. 


FIG.  126. 


The  form  shown  in  Fig.  124  (b)  is  most  common.  Here  the  bush- 
ing B  is  pressed  into  the  inner  links,  and  the  pin,  which  is  riveted 
fast  to  the  outer  links,  bears  over  the  whole  length  of  the  bushing. 
The  roller  R  rotates  on  the  bushing.  In  the  block  chain,  Fig. 
124  (c),  the  pin  also  bears  over  the  whole  thickness  of  the  block 
D,  but  since  the  roller  is  necessarily  omitted,  there  is  more  friction 
against  the  tooth.  Roller  chains  may  be  used  for  velocities  up 
to  about  800  feet  per  minute,  and  block  chains  up  to  about  500 
feet  per  minute. 

The  defect  in  the  operation  of  the  roller  or  block  chain  may 
be  seen  by  referring  to  Figs.  125  and  126.  When  the  chain  is 
new,  and  has  the  same  pitch  as  the  wheel,  it  fits  down  on  the  wheel 
as  shown  in  Fig.  125,  but  in  a  very  short  time  the  chain  stretches 


346 


MACHINE  DESIGN 


slightly,  due  to  wear  of  the  joints,  thus  increasing  the  pitch  of 
the  links.  The  wheel,  on  the  other  hand,  may  wear  but  this  does 
not  change  the  pitch.  The  operation  of  the  chain  is  then  as  shown 
in  Fig.  126,  the  increased  pitch  causing  the  rollers  to  ride  higher 
and  higher  on  the  back  of  the  tooth  as  they  move  round  the 
sprocket.  The  roller  A  is  shown  fully  seated  while  B  is  just 
coming  down  to  its  seat.  Before  B  can  become  fully  seated  A 
must  rise,  and  this  action  takes  place  when  A  and  B  are  carrying 
full  load.  As  a  consequence  the  chain  does  not  run  quietly  and 
smoothly  and  the  wear  is  excessive,  thus  limiting  the  speed  at 
which  the  chain  may  be  run.  This  difficulty  is  sometimes  met 
by  the  arrangement  shown  in  Fig.  127.  Here  the  pitch  of  the 


FIG.  127. 

chain  when  new  is  made  a  little  less  than  the  pitch  of  the  driving 
sprocket,  and  clearance  is  allowed  between  the  roller  and  the 
tooth,  so  that  the  driving  is  done  by  the  last  tooth  L;  the  pitch 
of  the  chain  being  such  that  the  incoming  roller  M  just  clears  the 
back  of  the  first  tooth  and  seats  itself  close  to  it  at  the  root  as  at 
N.  As  the  chain  stretches,  the  rollers  move  backward  toward 
the  faces  of  the  teeth,  till  a  condition  like  that  in  Fig.  125  is  reached, 
and  riding  commences.  The  pitch  of  the  driven  sprocket  wheel 
is  made  equal  to  that  of  the  chain,  and  the  condition  when  new 
is  that  shown  in  Fig.  127.  As  the  chain  stretches,  the  rollers 
move  gradually  backward  away  from  the  driving  faces  of  the 
tooth,  the  driving  being  done  on  the  last  tooth  P.  It  is  evident 


BELT,    ROPE,    AND    CHAIN    TRANSMISSION  347 

that  this  construction  extends  the  time  preceding  the  condition 
shown  in  Fig.  126.  When  this  construction  is  used,  the  form  of 
the  tooth  must  be  slightly  modified.  Referring  to  Fig.  128  it  is 
obvious  that  if  the  outline  of  the  tooth  M  be  an  arc  of  a  circle 
struck  from  the  centre  of  the  roller  (2),  this  roller  will  swing  from 
its  position  i'  rolling  on  the  face  of  the  tooth,  and  this  is  the 
usual  outline.  But  before  roller  (3)  can  take  the  load,  which  (2) 
is  about  to  give  up,  it  must  be  fully  rooted  against  the  next  tooth; 
whereas  (from  Fig.  128),  a  small  distance  now  separates  the  two. 
Therefore  as  (2)  rolls  up  the  curve  of  the  tooth  it  should  allow 
(3)  to  slowly  settle  back  in  place.  The  tooth  outline  is  therefore 
struck  (as  shown  on  M),  from  a  point  a  little  inside  the  pitch 
polygon  so  as  to  give  a  curve  tangent  to  the  first  and  last  positions 


FIG.  128. 

of  the  roller.  This  outline  is  also  necessary  for  the  back  of  the 
tooth  in  order  to  allow  the  incoming  roller  to  swing  in  without 
striking.  The  velocity  of  the  chain  is,  therefore,  a  little  less  than 
the  theoretical  velocity  on  account  of  this  continual  slipping  back- 
ward. Brief  reflection  will  show  that  the  tooth  outlines  of  the 
driven  sprocket  may  be  struck  from  the  centre  of  the  roller  when 
rooted  in  place;  and  that  when  the  chain  is  stretched  a  little  it 
will  creep  as  it  is  wound  upon  the  driven  sprocket. 

When  the  roller  (4),  Fig.  127,  is  about  to  roll  up  the  face  of 
P,  roller  (5)  is  not  in  contact  with  M  (wear  having  begun) ; 
hence  the  chain  will  move  ahead  till  (5)  is  in  full  contact  with  M. 

The  greatest  defect  in  this  construction  is  the  fact  that  the 
load  is  carried  entirely  on  one  tooth  and  hence  the  wear  is  ex- 


348 


MACHINE   DESIGN 


cessive.  This  may  be  so  great  that  the  chain  creeps  forward  on 
the  driven  wheel  so  as  to  cause  the  incoming  roller  to  strike  the 
tooth  S,  Fig.  127. 

The  above  difficulties  are  overcome  in  the  so-called  silent 
chains.  In  these  chains  the  inevitable  stretching  of  the  links 
is  compensated  for  in  a  peculiar  manner.  The  true  theory  of  the 
action  of  these  chains  is  very  complex;  but  the  general  action  is 
as  follows: — as  the  chain  stretches,  the  links  continually  tend 
to  take  up  a  position  farther  and  farther  away  from  the  centre  of 
the  sprocket;  thus  increasing  the  length  of  the  sides  of  the  pitch 
polygon  to  suit  the  elongation  of  the  link.  Each  link  therefore 
remains  in  constant  contact  with  its  own  tooth,  from  the  time  of 


FIG.  129. 

engagement  till  release  takes  place.     The  links  seat  themselves 
without  sliding  action  and  the  operation  is  nearly  noiseless. 

In  the  Renold  chain  of  this  type,  Fig.  129,  the  links  move 
relative  to  each  other  on  a  round  pin  P,  the  shouldered  ends  of 
which  are  riveted  into  a  washer  W,  thus  holding  the  chain  together. 
In  a  later  form  half  bushings  of  bronze  are  so  fitted  to  the  links 
that  the  pin  has  a  bearing  over  its  full  length;  but  the  relative 
motion  of  the  pin  to  the  bush  is  still  a  sliding  motion.  In  the 
Morse  chain  this  sliding  is  eliminated  by  an  ingenious  form  of 
rocker  joint  shown  in  Fig.  130.  The  hardened  steel  parts,  A 
and  B,  are  fitted  respectively  to  the  sets  of  links,  D  and  C.  While 
keeping  contact  along  a  fixed  line  they  rock  on  each  other  as  the 
links  C  and  D  move  relatively  to  each  other,  and  sliding  is  thus 


BELT,    ROPE,    AND   CHAIN    TRANSMISSION 


349 


eliminated.  When  transmitting  simple  tension  between  the 
sprockets,  the  parts  A  and  B  are  in  contact  on  flat  surfaces  as 
shown  at  E.  This  construction  has  the  advantage  of  requiring 
little  or  no  lubrication,  hence  the  chain  maybe  run  at  higher  speeds 
than  others  requiring  lubrication,  the  speeds  of  which  are  limited 
by  the  velocity  at  which  centrifugal  action  throws  off  the  lubri- 
cant. The  Morse  chains  also  work  well  in  dusty  places. 

The  efficiency  of  both  of  these  chains  is  very  high,  the  makers 
of  the  Morse  chain  claiming  an  efficiency  of  nearly  99  per  cent. 


FIG.  130. 

Such  chains  are  particularly  useful  for  connecting  shafts  which  are 
too  far  apart  for  gearing,  and  not  far  enough  for  a  belt,  and  in 
places  where  positive  connection  is  desirable,  as  in  motors 
driving  heavy  machine  tools.  It  is  to  be  especially  noted  that 
this  form  of  transmission  requires  no  definite  tension  on  the 
slack  side  of  the  chain  to  produce  a  certain  driving  force  on 
the  tight  side;  and  hence  the  pressure  on  the  bearings  is  much 
reduced,  for  a  given  effective  pull  on  the  wheel  rim. 


CHAPTER    XIII 


APPLICATIONS   OF   FRICTION 

Friction  Wheels  for  Power  Transmission 

156.  General  Considerations.  When  it  is  required  to  drive 
a  rotating  member  intermittently,  and  the  rate  of  driving  is 
not  necessarily  positive,  friction  wheels  have  been  found  very 
useful.  They  are  particularly  applicable  where  the  amount  of 
power  is  comparatively  small,  as  in  feed  mechanisms,  but  they 
may  also  be  used  for  heavy  work  when  properly  constructed. 
For  continuous  driving  the  transverse  sections  of  friction  wheels 


FIG. 


FIG.  133. 


must  be  circular  in  cross-section,  and  this  form,  only,  is  used  in 
practice. 

Figures  131  and  132  show  common  forms  of  friction  wheels. 
In  Fig.  131  let  A  be  the  driving  wheel  which  rotates  continuously, 
and  let  B  be  the  driven  wheel  which  is  required  to  be  driven 
intermittently.  The  shaft  of  A  is  so  mounted  that,  by  means  of  a 
lever  attached  to  the  bearing,  A  may  be  pressed  up  against  B 
with  a  force  P,  or  it  can  be  moved  slightly  away  from  B  until  no 
contact  exists.  If  now  the  force  P  is  applied  to  the  bearing 
(which  should  be  close  to  A),  an  equal  and  opposite  force  is  set 
up  in  the  bearing  of  B,  and  the  wheels  are  pressed  together  at  the 


35° 


APPLICATIONS    OF    FRICTION 


351 


line  of  contact.  The  resistance  to  slipping  at  the  line  of  contact 
will  be  /*P,  where  /JL  is  the  coefficient  of  friction  of  the  materials 
of  which  the  wheels  are  made;  and  if  /J.P  is  equal  to,  or  greater 
than,  the  resisting  force  at  the  surface  of  B,  A  will  cause  B  to 
rotate.  Theoretically,  A  and  B  will  roll  together  with  pure  rolling 
motion,  but  practically  this  cannot  be  attaine^l,  as  even  with  very 
hard  materials  the  wheels  flatten  slightly  at  the  line  of  contact. 
(See  Art.  108.) 

Fig.  132  illustrates  the  application  of  friction  wheels  to  shafts 
which  are  not  parallel  to  each  other,  the  wheels  here  having  the 
form  of  rolling  cones.  Obviously  the  principle  is  of  wide  applica- 
tion and  many  combinations  of  friction  wheels  are  used.  Fig. 
133  illustrates  a  friction  wheel  arranged  so  that  the  driver  A 


FIG.  134. 


FIG.  135  (b). 


FIG.  135  (a). 


can  rotate  the  driven  wheel  B  in  either  direction,  depending  on 
whether  it  is  pressed  against  the  surface  m  or  the  surface  n. 

Fig.  134  shows  a  form  of  friction  mechanism  much  used  for 
imparting  variable  speed  to  the  driven  shaft.  The  driver  A 
may  be  moved  along  the  shaft  C  at  will.  When  at  A'  the  angular 
velocity  of  B  is  a  minimum.  As  A  is  moved  inward,  the  rotative 
velocity  of  B  increases.  When  A  is  moved  across  the  centre  of  B 
to  the  other  side,  the  direction  of  rotation  of  B  is  reversed.  If 
A  were  infinitely  thin,  it  would,  theoretically,  roll  upon  B  with 
pure  rolling  motion.  Since,  however,  it  must  have  an  appreciable 
width  of  face,  and  since  the  velocity  of  B  varies  with  the  radius, 
it  is  evident  that  there  must  be  some  sliding  at  the  line  of  contact. 
For  this  reason  the  thickness  of  A  must,  for  best  results,  be  kept 
small  compared  to  the  radius  of  B. 


352  MACHINE  DESIGN 

157.  Materials  for  Friction  Wheels.     The  driven  wheels  of 
friction  devices  should  always  be  made  of  a  harder  material  than 
the  driver,  for  the  reason  that  the  driven  wheel  is  likely  at  any 
time  to  be  held  stationary  by  the  load,  while  the  driving  wheel 
revolves  against  it  under  pressure.     This  action,  while   severe 
on  the  driver,  does  not  tend  to  wear  it  out  locally,  while  it  does 
rapidly  wear  flat  spots  on  the  driven  wheel.     Driven  wheels  are, 
therefore,  almost  universally  made  of  iron,  and  driving  wheels 
of  wood,    leather,    paper,    rubber  or   of    some    composition  of 
these;   the  most  common  being  leather  and    various  forms  of 
paper. 

158.  Practical  Coefficients.     The  tangential  force  F,  exerted 
by  A  upon  B,  Fig.  131,  is  dependent  on  the  pressure  P  and  the 
coefficient  of  friction  ^.     It  is,  therefore,  necessary  to  know  the 
allowable  pressure  per  unit  of  length  along  the  contact  elements 
and  also  the  value  of  /*  for  the  particular  materials  used.     The 
most  comprehensive  investigation  of  these  relations  is  that  made 
by  Professor  Goss,*  whose  experiments  cover  a  variety  of  materials. 
He  recommends  the  following  pressures,  which  are  about  one-fifth 
of  the  ultimate  crushing  strength   of    the  respective  materials. 

SAFE  WORKING  PRESSURES  PER  INCH  OF  CONTACT. 

Material.  Pressure. 

Straw  fibre 150 

Leather  fibre    240 

Tarred  fibre  240 

Leather       150 

Woodf 100  to  150 

Professor  Goss  found  that  the  coefficient  of  friction  for  all  the 
wheels  tested  approached  a  maximum  value  when  the  slip  be- 
tween the  two  wheels  was  about  2  per  cent,  and,  within  narrow 
limits,  was  practically  independent  of  the  pressure  of  contact.  He 
found  these  values  to  range  for  different  combinations  from  low 
values  up  to  .515.  In  these  experiments  the  friction  due  to  the 
bearings  was  neglected.  The  bearings,  however,  were  of  the 
roller  type  and,  probably,  absorbed  less  power  than  the  ordinary 

*  See  Transactions  A.  S.  M.  E.   Vol.  XXIX. 

f  The  value  for  wood  is  not  from  Professor  Goss's  paper. 


APPLICATIONS   OF    FRICTION  353 

bearing.  Making  due  allowance  for  the  difference  between 
laboratory  conditions  and  those  found  in  practice,  Professor 
Goss  recommends  the  following  approximate  values  of  ^  *  for 
the  various  combinations.  In  this  connection  it  is  to  be  noted 
that  allowance  must  be  made  for  a  decrease  in  the  value  of  this 
coefficient  when  the  linear  velocity  of  the  driver  is  great,  in 
the  case  where  the  driver  is  starting  the  driven  wheel  under 
load  (see  Art.  28). 

WORKING  VALUES  OF  COEFFICIENT  OF  FRICTION. 
Materials.  Coefficient  of  Friction. 

Straw  fibre  and  cast  iron 0.26 

Straw  fibre  and  aluminum 0.27 

Leather  'fibre  and  cast  iron     0.31 

Leather  fibre  and  aluminum     o .  30 

Tarred  fibre  and  cast  iron 0.15 

Tarred  fibre  and  aluminum o .  18 

Leather  and  cast  iron 0.14 

Leather  and  aluminum    .  . . 0.22 

Leather  and  typemetal     0.25 

Wood  and  metal 0.25 

159.  Power  Transmitted  by  Friction  Wheels.     If  V  be  the 

velocity  of  the  surface  of  the  friction  wheels  in  feet  per  minute, 
P  the  total  normal  pressure  in  pounds,  F  the  resulting  tangential 
force,  and  ta  the  coefficient  of  friction;  then  since  F  =  pP,  the 
rate  at  which  power  is  transmitted  in  foot  pounds  per  minute  is 
jj.P  V,  and  the  horse-power  is 

HP  =  ^- 00 

33,000 

of  if  d  be  the  diameter  of  the  driver  in  inches,  /  the  length  of  face 
in  inches,  w  the  allowable  load  per  inch  of  face,  and  AT"  the  number 
of  revolutions  per  minute,  the  horse-power  is 

H.P.  -  »*>lX«dN  I  o.oooooS^ldN.  (2) 

12  x  33>°°° 

Example.  How  many  horse-power  can  be  transmitted  by  a 
straw-fibre  friction  pulley  of  8"  diameter  and  6"  face,  when  run- 
ning at  500  r.p.m.,  the  driven  wheel  to  be  of  cast  iron? 

*  The  coefficient  for  wood  is  not  from  Professor  Goss'  paper. 
23 


354  MACHINE   DESIGN 

Here  d  =  8",  /  =  6",  N  =  500,  /*  =  0.26,  w  =  150 

.  '  .  H.  P.  =  .000008  X  0.26  X  150  X  6  X  8  X  500  =  7.5 

It  may  be  noted  that  the  horse-power  per  inch  of  width  of 
face  is  a  little  more  than  unity,  for  a  surface  speed  of  1,000  feet, 
as  in  the  above  example.  This  corresponds  closely  to  the  empiri- 
cal rule  given  for  belts  in  Art.  136;  and  corroborates  the  empirical 
rule  often  used  that  the  same  width  of  face  is  necessary  for  a 
friction  wheel  as  for  a  belt,  to  transmit  a  given  horse-power  at 
the  given  speed. 

In  the  case  of  bevel  wheels  (see  Fig.  132)  the  component  R 

p 

of  the  applied  force  P  presses  the  wheels  together  and  R  = 

The  velocity  of  the  mean  circumference  of  the  driver  may  be 
taken  as  the  velocity  of  transmission. 

In  face  friction  driving  as  in  Fig.  134,  the  width  of  the  driving 
wheels  should  be  kept  as  narrow  as  possible  for  best  results. 
If  the  velocity  of  the  outer  edge  of  the  driving  wheel  is  not 
more  than  4  per  cent  greater  than  that  of  the  inner  edge,  the 
above  coefficients  may  be  used.  Where  the  driver  must,  at  times, 
drive  at  a  short  distance  from  the  centre,  lower  values  of  the 
coefficient  of  friction  must  be  taken. 

The  faces  of  a  pair  of  metal  friction  wheels  are  sometimes 
formed  as  shown  in  Fig.  135  (a),  and  are  then  known  as  wedge- 
faced  friction  wheels.  The  object  of  this  construction  is  to 
secure  a  greater  resistance  to  slipping,  with  a  given  radial  pressure. 
It  is  to  be  noted  that  the  number  of  wedges  does  not  affect  this 
ratio,  but  decreases  the  wear  by  distributing  it  over  several  sur- 
faces. This  last  item  is  important,  as  it  is  easily  seen  that  the 
contact  surfaces  of  the  driver  and  the  driven  wheel  can  have  the 
same  velocity  at  one  point  only,  and  that  at  all  other  points 
slipping  or  a  grinding  action  occurs  and  wear  must  result.*  The 
teeth  therefore  should  not  be  very  long. 

In  Fig.  135  (b),  let  P  be  the  radial  force  applied  to  the  wedged 

D 

surface,  F  the  tangential  force  transmitted,  -  :  the  reaction  on 

*  See  "Kinematics  of  Machinery,"  by  John  H.  Bnrr,  pige  10  . 


APPLICATIONS   OF   FRICTION  355 

each  face  and  20  the  angle  of  the  wedge;   then  the  wedge  is  held 

73 

in  equilibrium  by  the  force  P,  the  reactions  -  -  and  the  frictional 

r> 

resistances  //  —  due  to  the  wedging  action.     Equating  vertical 

forces 

P  =  2  (—sine  +  —  cos  o)  or  since  F  =  2  (—  }  or  R  =  - 

\    2  2  '  \    2     /  fi 

F  sin  0 
P  =  -  -  +  F  cos  0       .      .      ,      .     (i) 


.  .  .     . 

sin  0  +  }jt  cos  0 

To  avoid  sticking  the  angle  2  0  should  not  be  less  than  30°. 


FRICTION  BRAKES 

1 60.  Friction  brakes    are  used  for  controlling  and  stopping 
machinery  by  absorbing  energy  through  frictional  resistance  from 
some  moving  part,  and  dissipating  it  as  heat.     Brakes  used  in 
heavy  work,  and  as  dynamometers  for  measuring  energy,  must 
often  be  fitted  with  water  circulation  to  carry  away  the  heat. 
The  student  is  referred  to  treatises  on  power  measurement  for 
a  discussion  of  dynamometers. 

161.  Block  Brakes.     The  simplest  form  of  brake  is  the  block 
brake  as  shown  in  Fig.  136.     Here  the  force  P,  acting  on  the 
lever  A,  presses  the  block  C  against  the  wheel  B.     Let  the  reaction 
between  the  wheel  and  the  block  be  R.     Then  if  B  be  rotating,  a 
tangential  frictional  resistance  {J.R  =  F  will  oppose  its  motion. 
\Vith  the  arrangement  shown  in  Fig.  136,  the  line  of  action  of  F 
passes  through  O  the  centre  of  the  fulcrum  of  A.     Considering  A 
as  a  free  body  and  taking  moments  around  O,  then  for  rotation 

m  either  direction 

p 
p  (a  +  b)  =  R  b    or  since  R  =  — 


356 


MACHINE   DESIGN 


In  Fig.  137  the  line  of  action  of  F  does  not  pass  through  O  and 
therefore  in  writing  the  equation  for  the  equilibrium  of  A  its 
effect  must  be  considered,  whence 


The  minus  sign  is  to  be  used  for  rotation  in  a  clockwise 
direction,  for  the  arrangement  shown,  and  the  plus  sign  for  rota- 


FIG.  137. 


FIG.  138. 


tion    in  the  opposite  direction.     It  is  to  be  especially  noted  that 

/£  c,  P  =  o; 


for  clockwise  rotation  when  -  =  — ,  or  when  b 


that  is,  the  brake  is  self-acting  and  if  put  in  contact  the  moment 
of  the  frictional  force  will  apply  it  with  ever-increasing  pressure. 
Obviously  such   proportions  should  be  avoided. 
In  a  similar  manner  for  Fig.  138 

Fb 


a  +  b 


\    r±| 

L     —  b 


(3) 


the  plus  sign  referring  to  clockwise  rotation,  for  the  arrangement 
shown,  and  the  minus  sign  to  rotation  in  the  opposite  direction. 
In  this  class  of  brakes  the  pressure  of  the  brake  R  against 
the  wheel  is  opposed  by  an  equal  force  R'  at  the  bearing  near  the 
wheel.  In  the  calculations  above,  the  braking  effect  due  to 
friction  of  the  journal  is  neglected,  as  its  lever  arm  is,  usually, 
small.  It  cannot  be  neglected  in  designing  the  bearing,  and  for 
this  reason  this  form  of  brake  is  not  well  adapted  to  heavy  work. 


APPLICATIONS    OF    FRICTION 


357 


Fig.  139  shows  an  arrangement  of  brake  beams  for  heavy  work 
such  as  is  used  in  mining  machinery.  The  force  W,  which  may 
be  applied  by  a  steam  cylinder,  acting  on  the  system  of  levers, 
causes  the  brake  beams  B  and  B  to  press  equally  on  opposite  sides 
of  the  wheel,  and  causes  no  pressure  on  the  bearings  of  the  drum. 

T 

If  —  be  the  tension  in  each  of  the  rods  A  and  A ,  the  frictional  force 
2 

exerted  on  the  wheel  is  F  =  2  ft  T.  If  the  pin  O  is  so  located 
that  when  the  load  W  is  applied  it  moves  to  O',  and  the  centre 


w 


FIG.  139. 

line  of  the  rod  A  passes  through  the  centre  of  the  pin  P,  a  toggle 
effect  is  obtained  and  the  tension  in  the  rods  A  and  A  may  be  made 
any  desirable  value;  in  fact  with  such  an  arrangement  care  must 
be  exercised  in  adjusting  the  brake  that  such  pressures  are  not 
brought  on  the  pins  as  will  cause  failure  by  shearing.  When  O 
moves  down  to  O'  the  brake  is  " locked"  in  position  and  the 
operating  force  may  be  removed.  This  last  feature  is  often  a 
valuable  quality  in  a  brake.  Brakes  of  this  character  are  gener- 
ally lined  with  wooden  blocks  as  shown. 

162.  Strap  Brakes.     If  the  effect  of  centrifugal  force  is  neg- 
lected  (see  Art.  131),  and  the  total  tensions  in  the  band  (T1 


358 


MACHINE    DESIGN 


and  T2)  be  taken  instead  of  the  tensions  per  inch  of  width,  equa- 
tion (6)  of  that  article  reduces  to 

*i       ?\ 

7  =  j*  =  iok (i) 

h        1  2 

Where  &  =  0.0076  /*  a,  a  being  the  arc  of  contact  in  degrees.  If, 
also,  F  is  the  total  frictional  force  exerted  by  the  band  upon  the 
wheel, 

It  is  obvious  that  these  equations  are  applicable  to  the  discussion 
of  band  brakes.  Figs.  140,  141,  and  142  show  the  most  usual 
arrangement  of  band  brakes.  In  Fig.  140  the  end  of  the  strap 


FIG.  141. 


FIG.  142. 


which  is  subjected  to  the  greatest  tension  7\  is  anchored,  for  con- 
venience, at  the  pin  which  serves  as  a  fulcrum  for  the  operating 
lever  L;  it  could  be  anchored  to  any  other  convenient  part  of  the 
frame. 

From  (i)  and  (2),     T2  - 
Taking  moments  around  O 

p  a  =  T,  b  = 


10  —  i 


Fb 


10  —  i 
Fb 


TR^O     ••,•••    (3) 

which  expresses  the  relation  between  the  applied  force  P  and  the 
frictional  resistance  applied  to  the  wheel. 


APPLICATIONS    OF    FRICTION  359 

In  Fig.  141  the  end  under  greatest  tension  is  attached  to  the 
lever  and  the  end  of  least  tension  is  anchored,  hence  for  this  case 

Fb 


In  Fig.  142  the  end  under  greatest  tension  is  anchored  to  the  lever 
at  a  shorter  radius  than  the  end  of  least  tension;  so  that  the  force 
which  it  exerts  assists  the  operating  force  P.  This  is  known  as 
a  differential  brake.  For  this  case  in  a  similar  manner  as  above 

F_  r&.-io^-. 

'   a   I  iok-i  J  '      '  r  '      (5) 

It  is  to  be  especially  noted  that  if  iok  ^  =  b2,  P  =  o,  and 
the  band  will  brake  automatically;  that  is  if  any  force  is  applied 
to  the  lever,  the  brake  will  continue  to  set  itself  up  with  ever 
increasing  force  till  motion  ceases  or  rupture  occurs.  This  form 
of  brake  is  exceedingly  dangerous  on  account  of  its  tendency  to 
"grab/'  especially  if  /*  is  materially  increased  through  a  change 
in  the  character  of  the  friction*  surfaces. 

Strap  brakes  are  usually  made  of  wrought  iron  or  steel.  In 
light  work  they  may  engage  with  a  cast-iron  surface  or  may  be 
lined  with  leather;  but  in  very  heavy  work  they  should  be  lined 
with  wood. 

FRICTION  CLUTCHES  AND  FRICTION  PULLEYS 

163.  Friction  clutches  though  made  in  a  great  variety  of  forms 
can,  in  a  large  measure,  be  classified  under  four  principal  types, 
namely,  Conical,  Radially  Expanding,  Disc,  and  Band.     A  well- 
designed  clutch  should  start  its  load  quickly  but  without  shock, 
and  should  disengage  quickly.     It  should  be  "self-sustained," 
that  is,  when  the  clutch  is  driving,  no  external  force  should  be 
necessary  to  hold  the  contact  surfaces  together.     In  addition,  it 
is  often  necessary  that  the  clutch  should  "lock"  in  place,  after 
the  manner  of  the  brake  in  Fig.  139. 

164.  Conical  Clutches.     Fig.   143  shows  the  elements  of  a 
conical  clutch  which  is  self-sustained.     The  cone  F  is  fast  to  the 
shaft  S  and  rotates  with  it.     The  pulley  H  rotates  upon  F  and 


36o 


MACHINE    DESIGN 


carries  with  it  the  levers  E.  When  the  thimble  B  is  forced  under 
the  rollers  C,  the  levers  E  force  the  cone  surfaces  in  contact. 
Heavy  springs  at  G  (not  shown)  throw  the  surfaces  apart  when 
the  thimble  is  withdrawn.  The  relation  between  the  trans- 
mitted frictional  force  F  and  the  force  P  applied  to  the  cone,  in  a 
direction  parallel  to  the  axis,  is  the  same  as  that  of  the  wedge 
gearing  in  Art.  159,  or 

(6) 


F  = 


sin  0  +  V-  cos  Q 


The  angle  0  should  not  be  less  than  10°,  unless  some  mechanism 
is  provided  for  separating  the  cone  surfaces,  positively,  when 


FIG.  143. 


FIG.  144. 


desired.  For  clutches  that  do  not  operate  frequently,  metal 
surfaces  are  often  used;  but  where  the  operation  of  clutching  is 
frequent,  one  surface  is  usually  lined  with  wood,  cork,  or  leather. 
165.  Radially  Expanding  Clutches.  Fig.  144  shows  the  ele- 
ments of  a  radially  expanding,  self-sustained  clutch.  The  clutch 
body  A  is  keyed  to  the  shaft,  while  the  pulley  C  rotates  loosely 
upon  the  shaft.  The  circular  segment  B,  which  fits  the  inner 
surface  of  C,  can  be  moved  radially  upon  A .  The  loose  ring  G 
is  operated  axially  by  a  forked  lever  fitting  on  the  pins  P.  When 
the  sleeve  E  is  forced  inward  by  the  ring  G,  the  links  D  force  the 
•segments  B  outward  against  C.  In  the  arrangement  shown  the 
links  have  a  toggle  effect  and  can  exert  enormous  pressure  against 


APPLICATIONS    OF    FRICTION  361 

B,  hence  adjustment  must  be  carefully  performed.  This  is 
usually  accomplished  by  making  the  length  of  the  link  D  adjust- 
able, by  means  of  turn-buckles  or  similar  devices,  which  also 
provide  a  means  of  compensating  for  wear.  Usually  the  sleeve 
has  motion  enough  to  carry  the  inner  end  of  the  link  slightly  past 
the  centre  position  shown,  thus  locking  the  clutch  in  place. 

166.  Disc  Clutches.  Fig.  145  shows  the  elements  of  amultiple- 
disc  clutch  as  sometimes  used  in  automobile  work  for  connecting 
the  engine  to  the  transmission  shaft,  A  being  fast  to'  the  engine 
shaft  and  B  to  the  transmission  shaft.  The  part  A  carries  a 
number  of  discs,  C,  which  fit  loosely  in  a  radial  direction  but  are 
prevented  from  rotating  relatively  to  A  by  bolts  E  which  also 
hold  L,  the  cover  of  the  case,  in  place.  A  second  set  of  discs  D, 
placed  alternately  between  the  discs  C  are  carried  on  the  part  B 
and  compelled  to  rotate  with  it  by  the  keys  G.  A  heavy  helical 
spring  F  (sometimes  made  of  rectangular  section  as  shown) 
presses  the  two  sets  of  discs  together  with  a  known  load  P,  when 
the  clutch  is  in  and  the  shafts  connected.  The  sleeve  B  while 
compelled  by  the  feather  S  to  rotate  with  the  transmission  shaft 
A7,  can  be  moved  axially  by  means  of  the  grooved  collar  /  and  the 
ring  /;  /  being  made  fast  to  B  but  built  separately  from  it  for 
constructive  purposes  only.  When  B  is  moved  to  the  right  the 
spring  is  compressed  and  the  pressure  on  the  discs  relieved.  The 
discs  often  run  in  an  oil  bath  to  prevent  "grabbing."  It  is 
readily  seen  that  while  the  force,  P,  which  presses  each  pair  of 
contact  surfaces  together  is  the  same,  the  total  frictional  force 
transmitted  is  proportional  to  the  number  of  pairs  of  contact 
surfaces  n  or 

F  =  ^nP       .     v     ,  •-..     ,     .    '(7) 

If  the  mean  friction  radius  of  the  discs  be  r,  the  frictional  mo- 
ment transmitted  is  Fr  =  junPr.  In  Fig.  145,  n  =  7.  The 
above  form  of  clutch  is  known  as  the  Weston  clutch.  Obviously 
any  number  of  pairs  of  discs  may  be  used.  For  large  work  the 
discs  are  sometimes  made  of  iron  and  wood  (or  wood-faced). 
For  small  work,  alternate  discs  of  steel  and  brass  are  employed. 
Many  pairs  of  contact  surfaces  are  then  used  and  the  discs  run  in 


362 


MACHINE    DESIGN 


oil  to  prerent  "grabbing."  The  width  of  the  wearing  faces  of 
the  discs  should  be  made  small  to  prevent  undue  wear  toward  the 
outer  edges  of  the  discs  D,  as  in  a  thrust  block  (Art.  105).  It  is 
better  to  use  more  discs  of  a  smaller  diameter  than  a  few  of  great 
face. 

167.  Band  Clutches.  Fig.  146  illustrates  the  elements  of  a 
band  clutch.  The  clutch  wheel  A  (which  may  be  fast  to  one 
shaft)  carries  the  wood-lined  band  C.  When  the  thimble  F 
(which  slides  on  the  shaft)  is  forced  under  the  lever  E,  the  iron 
band  C  is  tightened  and  clutches  the  rim  of  the  driven  wheel  B. 


FIG.  145. 


FIG.  146. 


Obviously  the  principles  involved  are  identical  with  those  of  the 
strap  brake,  Fig.  140  of  Art.  162.  For  light  work  the  band  may 
be  lined  with  leather,  but  in  heavy  work,  such  as  mine  hoisting, 
blocks  of  bass  wood,  or  other  soft  wood,  are  used.  The  wood 
lining  is  usually  made  fast  to  the  strap,  though  occasionally  on 
very  large  diameters  they  are  attached  to  the  wheel  so  that  they 
may  be  turned  true  in  place.  These  clutches  are  made  self- 
locking  by  arranging  for  a  toggle  effect  in  some  one  of  the  operat- 
ing levers. 

Occasionally  the  band  is  made  to  expand  inside  of  the  rim  of 
the  wheel  to  be  driven.  It  is  to  be  noted  that  this  case  is  not 
the  same  as  the  one  just  discussed,  but  is  a  special  case  of  a 


APPLICATIONS    OF   FRICTION  363 

radially  expanding  clutch.  The  outward  force  exerted  by  the 
band  may  be  computed  by  the  theory  of  Art.  78,  considering 
the  band  as  a  thin  cylinder  under  compression,  the  compressive 
stress  at  any  section  being  that  due  to  the  pressure  applied  by  the 
operating  lever. 

168.  Magnetic  Clutches.  A  number  of  clutches  have  recently 
appeared  which  are  operated  magnetically.  These  are  most 
generally  of  the  disc  type.  Evidently  the  general  principles  above, 
regarding  transmissive  power,  apply  also  to  these  clutches.  In 
magnetic  brakes,  the  load  is  usually  applied  by  a  spring,  or 
weight,  and  released  by  magnetic  action,  thus  insuring  safety 
against  accident  should  the  electric  service  fail. 

Practical  Coefficients  for  Brakes  and  Clutches.  The  most 
usual  combinations  of  friction  surfaces  for  brakes  and  clutches 
are  wood,  leather,  or  cork  with  iron;  and  iron  with  iron.  In  the 
multiple-disc  type,  brass  or  bronze  on  iron  or  steel  are  sometimes 
used.  Mr.  C.  W.  Hunt  gives  the  following  values  of  /*  as  the 
result  of  considerable  experience  in  designing  clutches,  namely: 
cork  on  iron,  0.35;  leather  on  iron,  0.3;  and  for  wood  on  iron  0.2. 
For  iron  on  iron  //  may  be  taken  as  0.25  to  0.3.  It  should  be 
remembered  that  if  the  friction  surfaces  are  to  be  engaged  at 
high  velocity,  lower  values  must  be  assumed  than  for  lower 
speeds  (see  Art.  28) . 

The  pressure  per  unit  area  of  surface  is  also  an  important 
feature  in  the  design  of  friction  machinery,  for  if  this  is  taken 
too  high,  excessive  wear  will  result.  Thus  in  disc  clutches  the 
pressure  is  usually  taken  at  not  more  than  25  to  30  pounds  per 
square  inch  and  lower  values  are  desirable.  Wooden  surfaces 
should  not  be  loaded  beyond  20  to  25  pounds  per  square  inch. 
If  the  clutch  or  brake  is  to  operate  frequently,  ample  surface 
must  be  provided  to  properly  radiate  the  heat  generated. 

References: — 

Transactions  A.  S.  M.  E.,  Vol.  XXX,  1908. 
Transactions  Inst.  Mechanical  Engineers,  July,  1903. 


CHAPTER  XIV 
TOOTHED  GEARING 

169.  General  Principles.  When  it  is  necessary  that  rotation 
of  one  shaft  shall  produce  definite  and  positive  rotation  of  another, 
it  is  evident  that  friction  wheels,  as  discussed  in  the  preceding 
chapter,  will  not  suffice  where  any  considerable  amount  of  power 
is  to  be  transmitted.  In  such  cases  the  peripheral  surfaces  of 
the  transmission  wheels  are  provided  with  teeth,  so  that  the 
motion  shall  be  positive.  It  is  evident  that  any  pair  of  surfaces 
which  will  roll  together  with  pure  rolling  motion,  so  as  to  give  the 
required  velocity  ratio,  may  serve  as  a  basis  for  the  design  of  a 
pair  of  toothed  gears;  and  works  on  mechanism  treat  fully  of 
the  methods  of  drawing  the  sections  of  such  surfaces  for  various 
conditions  and  velocity  ratios.  Whether  the  elements  of  the 
surface  thus  outlined  shall  be  parallel  or  otherwise  will  depend  on 
the  angle  which  the  shafts  make  with  each  other,  as  in  the  case  of 
friction  wheels,  and  tooth  gearing  may  be  classified  *  according  to 
the  character  of  the  pitch  surfaces,  and  the  relation  of  the  axes, 
thus: 


Kind. 

Relation  of  Axes. 

Pitch  Surfaces. 

Spur 
Bevel 
Screw 
Skew 
.       Twisted 
Face 

Parallel 
Intersecting 
Not  in  one  plane 
Not  in  one  plane 
Any 
Any 

Cylinders 
Cones 
Cylinders 
Hyperboloids 
Any  of  the  above 
None,  strictly 

The  most  important  of  these  are  spur,  bevel,  and  a  few  special 
forms  of  twisted  and  screw  gears.  The  motion  transmitted  by  a 
pair  of  properly  designed  toothed  gears  is  identical  with  that 
of  the  base  curves  or  surfaces  rolling  together.  If  r t  and  r2  be 


*  See  "Kinematics  of  Machinery,"  by  John  H.  Barr,  page  no. 
364 


TOOTHED   GEARING 


365 


the  instantaneous  radii  of  such  a  pair  of  surfaces  at  the  point  of 
contact,  and  ^  and  ^2  be  their  instantaneous  angular  velocities, 

then  —  =  — .     In  the  most  common  case  the  angular  velocity 

^2       ri 

of  both  shafts  is  constant  and  hence  ^  and  r2  are  constant,  and 
the  rolling  surfaces  are  circular  in  cross-section.  Thus  Fig.  147 
shows  a  portion  of  two  gears  whose  rolling  surfaces  are  a  pair  of 
circular  cylinders,  represented  in  cross-section  by  the  circles  C 
and  D.  If  the  teeth  are  properly  proportioned  the  motion  trans- 
mitted will  be  identical  with  that  produced  by  the  rolling  of  C  on 
D.  It  can  be  shown  that  the  condition  which  such  tooth  outlines 


FIG.  147. 


FIG.  148. 


must  fulfil  in  order  that  the  velocity  ratio  may  be  constant,  is 
that  the  common  normal  to  the  tooth  outlines  at  the  point  of  contact 
must  always  pass  through  the  point  of  tangency  of  the  rolling 
circles.  There  are  many  curves  which  can  be  used  for  tooth 
outlines,  and  which  would  fulfil  the  condition,  but  in  practice 
only  two  are  commonly  employed,  namely,  the  involute  and  the 
cycloid. 

Fig.  147  illustrates  a  portion  of  two  gears  with  involute  teeth. 
The  upper  wheel,  M,  is  the  driver.  Contact  between  two  teeth 
has  just  begun  at  a,  and  the  common  normal  to  the  point  of  con- 
tact a  O  b  passes  through  the  pitch  point  O.  As  the  wheels  rotate 
the  point  of  contact  will  move  along  the  line  a  O  b  till  contact 
ceases  at  b.  Hence  in  the  involute  system  the  normal  to  the 


366  MACHINE  DESIGN 

point  of  contact  makes  a  fixed  angle  with  the  common  tangent  to 
the  pitch  circles. 

Fig.  148  shows  a  portion  of  two  gears  with  cycloidal  teeth. 
Contact  is  just  beginning  at  a,  and  as  the  gears  rotate  the  point  of 
contact  will  move  along  the  curved  path  a  O  b,  contact  ceasing  at 
b.  The  normal  to  the  first  point  of  contact  is  drawn,  and  it  is 
clear  that  the  inclination  of  the  normal  to  the  common  tangent 
of  the  pitch  circles,  is  a  maximum  at  this  point,  and  continually 
varies  in  direction  though  always  passing  through  the  point  O. 
It  can  be  shown  that  in  the  involute  system  the  angular  velocity 
ratio  will  remain  constant,  within  the  limits  of  action,  whether 
the  pitch  circles  are  tangent  or  not ;  but  for  the  transmission  of 
constant  velocity  ratio  with  cycloidal  gearing  the  pitch  circles 
must  remain  tangent.  The  involute  gear,  therefore,  has  a  decided 
advantage  for  general  use  and  it  has  practically  superseded  the 
cycloidal  for  most  work.  A  fuller  treatment  of  the  theory  of 
gear- tooth  outlines,  which  is  beyond  the  scope  of  this  work,  will 
be  found  in  treatises  on  mechanism.* 

170.  Interchangeable  Systems  of.  Gear  ing:  Standard  Forms. 
It  is  desirable  in  practical  work  that  any  gear  of  a  given  pitch 
shall  run  properly  with  any  other  gear  of  the  same  pitch.  In 
order  that  this  may  be  so,  certain  limitations  must  be  placed 
upon  the  form  and  dimensions  of  the  tooth.  In  the  cycloidal 
system  interchangeability  may  be  accomplished,  as  far  as  the 
tooth  outlines  are  concerned,  by  keeping  the  diameter  of  the 
describing  circle  constant  for  all  gears  of  the  series. 

Any  involute  tooth  outline  will  run  properly  with  any  other 
similar  outline;  and  any  gear  with  involute  teeth  will  run  with 
any  other  gear  having  similar  teeth,  as  far  as  the  length  of  the 
involute  outlines  will  permit,  providing  the  thickness  of  teeth 
will  allow  them  to  mesh.  In  order  to  obtain  involute  outlines  of 
sufficient  length,  and  a  series  of  gears  with  fixed  nominal  pitch 
circles,  the  angle  0,  Fig.  147,  made  by  the  line  of  action  with  the 
common  tangent  to  the  pitch  circles  must  have  a  proper  value, 
and  be  constant  for  all  gears  of  the  series.  In  the  systems  in 

*  See  "Kinematics  of  Machinery,"  by  J.  H.  Barr,  page  in;  also,  "  Machine 
Design,"  part  i,  by  F.  R.  Jones. 


TOOTHED  GEARING  367 


most  common  use  this  angle  is  14%°,  though  there  is  a  ten- 
dency in  modern  work  toward  a  greater  angle. 

It  is  found  undesirable  in  practice  to  make  gears  with  less  than 
twelve  teeth;  and  in  some  cycloidal  systems  the  radius  of  a 
twelve-tooth  gear  of  the  required  pitch  is  taken  as  the  diameter 
of  the  describing  circle.  For  a  twelve-tooth  gear  this  will  re- 
sult in  radial  lines  for  the  tooth  outlines  below  the  pitch  circle, 
i.e.,  the  tooth  will  have  radial  flanks.  In  the  practice  of  the 
Brown  &  Sharpe  Mfg.  Co.,  the  diameter  of  the  describing  circle 
is  the  radius  of  the  fifteen-tooth  gear  of  the  series.  This  gives 
spaces  between  the  flanks  of  the  teeth  on  the  twelve-  tooth,  or 
smallest  gear,  so  nearly  parallel  that  they  may  be  cut  with  a 
rotary  cutter. 

It  is  evident  from  Figs.  147  and  148  that  the  tooth  outlines  of 
any  system  may  be  extended  both  above  and  below  the  pitch  line 
till  they  meet.  It  is  also  clear  that  the  longer  the  teeth  the  earlier 
will  they  engage  with  each  other,  the  greater  will  be  the  arc  of 
contact,  and  the  greater  will  be  the  number  of  teeth  continually  in 
contact.  The  distribution  of  the  load  over  a  number  of  pairs 
of  teeth  is  in  itself  conducive  to  smooth  running;  but  on  the  other 
hand,  extending  the  arc  of  contact  away  from  the  pitch  point, 
increases  the  sliding  between  teeth,  and  also  the  velocity  with 
which  the  teeth  approach  each  other.  The  tooth  also  becomes 
weaker  as  it  is  lengthened,  the  thickness  remaining  the  same, 
and  for  these  reasons  a  practical  limit  is  placed  on  the  length  of 
teeth.  The  length  of  tooth  adopted  in  practice  is,  therefore,  a 
compromise  between  conflicting  conditions,  which  experience 
has  shown  will  give  good  results. 

The  distance  along  the  pitch  line  from  any  point  on  a  tooth  to 
a  corresponding  point  on  the  next  tooth,  is  called  the  circular 
pitch  ;  and  will  be  noted  by  s.  The  thickness  of  the  tooth  along 
the  pitch  line  will  be  denoted  by  t,  Fig.  151.  In  the  case  of  cut 

gears,  where  no  clearance  is  allowed  between  teeth,  /  =  -.      In 

some  forms  of  gears,  such  as  shown  in  Fig.  150,  where  a  metal 
pinion  engages  with  a  gear  having  wooden  teeth,  the  pitch  may 
not  be  equally  divided,  but  the  metal  tooth  may  be  thinner  than 


368  MACHINE  DESIGN 

the  wooden  tooth.  If  N  be  the  number  of  teeth  and  D  the 
diameter,  then  evidently  Ns  =  xD.  If  the  number  of  teeth  N 
be  divided  by  the  diameter,  the  quotient,  or  the  teeth  per  inch 
of  diameter,  is  called  the  diametral  pitch  and  will  be  denoted  by 

S.  Since  S  =  —  and  s  =  — -,  S  X  s  =  TT  .-.£  =  -  and 
^  =  -.  The  diametral  pitch  is,  ordinarily,  the  most  convenient 

o 

for  use,  and  in  this  country  practically  all  interchangeable  sys- 
tems are  based  upon  the  diametral  pitch.  Thus  a  gear  24"  in 
diameter  and  3  diametral  pitch  would  have  24  X  3  =  72  teeth, 

and  the  circular  pitch  would  be  —  =  1.05  inches.     In  the  sys- 

o 

tern  of  teeth  adopted  by  the  Brown    &  Sharpe  Mfg.  Co.,  and 
which  is  used  very  extensively  in  America,   the  following  pro- 
portions are  given  for  cut  teeth.     See  Fig.  151. 
Let    Z>j  =  the  outside  diameter  of  the  gear. 
"     D    =  the  pitch  diameter  of  the  gear. 
"     D2  =  the  diameter  of  a  circle  through  bottom  of  space. 
"     S     =  the  diametral  pitch. 
"     s     =  the  circular  pitch. 

"     a     =  the  addendum  =  height  of  tooth  above  pitch  line. 
"     c     =  the  clearance  between  top  of  tooth  and  bottom  of 

space  when  gears  are  in  mesh. 
"     d     =  the  dedendum,  or  total  depth  of  space  below  pitch 

line. 
"     t      =  the  thickness  of  tooth  on  pitch  line  =  width  of 

space  on  pitch  line  in  cut  teeth. 
"     N   =  the  number  of  teeth  in  gear. 
"     h     =  the  total  height  of  tooth. 

Then  N  =  D  S  =  — , 

~   2    ~   2~5 

/  7T 

*• ix  . 

10  ~~    20  S 


TOOTHED   GEARING  369 


d  =  a  +  c 
h  =  2  a  +  c 


and  D2  =  D  -  2  (a  +  c) 


x 

In  the  case  of  rough  gear  teeth,  cast  from  a  wooden  pattern,  the 
thickness  of  the  tooth  must  be  less  than  the  width  of  the  space,* 
and  the  clearance  at  the  bottom  of  the  space  must  be  greater  than 
in  cut  teeth.  If  the  gears  are  machine-moulded,  the  difference 
need  not  be  quite  so  great  as  in  pattern-moulded  gears.  For 
pattern-moulded  gears  good  practice  gives  t  =  0.45  s  for  large 
gears,  to  0.47  5  for  small  gears,  and  the  corresponding  width  of 
the  space  would  be  0.55  s  to  0.53  s.  For  machine-moulded  gears 
t  =  0.465  to  0.48  5  and  the  corresponding  space  would  be  0.54  s 
to  0.52  s. 

Table  XXVI  gives  dimensions  of  gear  teeth  for  cut  spur  gears, 
in  accordance  with  the  standards  of  the  Brown  &  Sharpe  Mfg.  Co. 

171.  Methods  of  Making  Gear  Teeth.  Metallic  gear  wheels 
are  either  cast  from  a  pattern,  or  the  rim  is  cast  or  forged  solid, 
and  the  teeth  are  cut  from  the  solid  metal  by  rotary  or  recipro- 
cating cutters.  Where  the  gear  teeth  are  cast,  it  is  very  important 
that  the  pattern  itself  be  very  accurately  made;  for  even  with  the 
greatest  care  in  moulding,  it  is  impossible  to  obtain  true  spacing, 
on  account  of  shrinkage  and  displacement  due  to  "rapping"  the 
pattern  in  the  sand.  For  this  reason,  and  on  account  of  the 
difficulty  of  obtaining  smooth  surfaces,  greater  clearance  must  be 
allowed  in  cast  gears  than  in  cut  gears,  as  already  noted.  Wooden 
patterns  are  very  unreliable  for  such  work,  on  account  of  their 
tendency  to  warp  and  shrink,  and  permanent  patterns  should  be 
made  of  metal.  If  the  pattern  for  a  spur  gear  is  withdrawn  from 
the  sand  with  a  movement  parallel  to  the  length  of  the  tooth,  the 
tooth  pattern  must  have  draft,  or  be  slightly  tapering  to  facilitate 
drawing,  and  consequently  the  cast  tooth  must  also  be  tapering. 
Care  should  be  taken  in  assembling  such  gears,  that  the  tapers  in 

*  The  difference  between  the  thickness  of  the  tooth  and  the  width  of  the  space 
is  commonly  called  "  backlash." 
24 


MACHINE  DESIGN 


the  two  gears  are  reversed  to  avoid  having  the  thick  ends  of 
both  sets  of  teeth  come  together,  thus  concentrating  the  pressure 
at  one  end.  Rough  cast  gears,  of  the  kind  described  above,  are 
used  only  for  rough  or  large  work,  and  not  for  high  speed.  The 
particular  defect  of  spur  gears  due  to  draft  does  not  exist  in  bevel 
gearing. 

In  gear-moulding  machines  the  pattern  consists  of  a  segment 
of  the   gear   pattern,   carrying  several   teeth.     The   pattern  is 

TABLE  XXVI 

PROPORTIONS    OF   GEAR   TEETH 


Diametral 
Pitch. 

S 

Circular 
Pitch. 

s    ~ 

Thickness  of 
Tooth. 

t 

Addendum 

£•".* 

a 

Depth  of 
Space  Below 
Pitch  Line. 

a  +  c 

Total  Depth 
of  Tooth. 

2  a  +  c 

I 

3.1416 

1.5708 

I  .0000 

I-I57I 

2-I57I 

li 

2-5*33 

I  .2566 

.8000 

•9257 

I-7257 

if 

2.0944 

I  .0472 

.6666 

•77*4 

1.4381 

if 

i  -7952 

.8976 

•57*4 

.6612 

i  .2326 

2 

1.5708 

•7854 

.5000 

•5785 

i  .078^ 

2-i 

1-3963 

.6981 

•4444 

•5r43 

•9587 

2\ 

i  .2566 

.6283 

.4000 

.4628 

.8628 

2  f 

1.1424 

•5712 

•3636 

.4208 

.7844 

3 

1.0472 

-5236 

•3333 

•3857 

.7190 

3i 

.8976  . 

.4488 

•2857 

•33o6 

.6163 

4 

•7854 

•3927 

.2500 

.2893 

•5393 

5 

.6283 

.3142 

.2000 

•2314 

•43*4 

6 

•5236 

.2618 

.1666 

.1928 

•3595 

7 

.4488 

.2244 

.1429 

•1653 

.3082 

8 

•3927 

.1963 

.1250 

.1446 

.2696 

9 

•3491 

•J745 

.HIT 

.1286 

•2397 

10 

.3142 

•i57i 

.IOOO 

•Ir57 

•2157 

ii 

.2856 

.1428 

.0909 

.1052 

.1961 

12 

.2618 

.1309 

•0833 

.0964 

•1798 

I3 

.2417 

.1208 

.0769 

.0890 

.1659 

14 

.2244 

.1122 

.0714 

.0826 

•i54i 

15 

.  2094 

.1047 

.0666 

.0771 

•1437 

16 

.1963 

.0982 

.0625 

.0723 

.1348 

17 

.  1848 

.0924 

.0588 

.0681 

.1269 

18 

•J745 

.0873 

•0555 

.0643 

.1198 

iQ 

•1653 

.0827 

.0526 

.0609 

•1135 

20 

•J571 

.0785 

.0500 

•°579 

.1079 

mounted  on  an  axis  in  such  a  manner  that  it  can  be  rotated 
accurately  through  any  portion  of  a  complete  revolution,  or 
" indexed."  In  forming  the  mould  the  segmental  pattern  is 
placed  in  position  and  sand  is  rammed  around  it.  The  pattern 
is  then  withdrawn  radially  and  rotated  to  the  next  succeeding 


TOOTHED   GEARING  371 

position  (the  indexing  device  insuring  accurate  spacing),  the 
operation  being  repeated  till  the  whole  circumference  is  moulded. 
The  mould  for  the  hub  and  arms  is  then  completed,  in  large  work 
this  last  being  often  accomplished  by  means  of  cores.  If  machine 
moulding  is  well  done  the  results  are  far  superior  to  those  obtained 
by  pattern  moulding,  and  gears  may  be  made  that  can  be  run  at 
moderately  high  speeds.  Obviously,  however,  all  cast  gears  are 
much  more  inaccurate  than  cut  gears,  and  the  latter  are  preferable 
where  high  speeds  and  smoothness  of  action  are  required. 

Metallic  gearing,  even  when  accurately  cut  and  aligned,  is 
inclined  to  be  very  noisy  when  run  at  a  peripheral  speed  of  more 
than  1,200  feet  per  minute,  especially  if  any  appreciable  "  back- 
lash" exists.  Relieving  the  points  of  the  teeth,  slightly,  reduces 
the  tendency  to  produce  noise.  Where  high  speeds  are  unavoid- 
able the  teeth  of  one  of  the  mating  gears  is  sometimes  made  of 
wood  or  rawhide.  Wheels  with  wooden  teeth  are  known  as 
mortise  wheels.  They  are  not  as  much  used  as  formerly,  because 
modern  methods  of  gear-cutting  produce  metallic  gears  of  such 
accurate  form  that  they  may  be  run  in  places  where  mortise  gears 
were  formerly  considered  indispensable.  In  making  mortise 
wheels  the  wooden  teeth  are  roughed  out  and  the  shank  is  fitted 
into  openings  cast  in  the  rim  of  the  wheel,  as  shown  in  Figs.  149 
and  150.  The  teeth  are  held  in  place  by  the  keys,  K,  or  pins,  P, 


as  shown.  The  teeth  proper  are  dressed  to  correct  form  with 
hand  tools  or  by  special  machines  using  a  fine  circular  saw  for  a 
cutter. 

Usually  the  large  gear,  only,  is  made  with  wooden  or  "  mortise" 
teeth,  the  pinion  being  made  of  metal.  This  is  rational  since 
the  pinion,  on  account  of  the  shape  of  its  teeth,  is  the  weaker  of 


372  MACHINE   DESIGN 

the  two,  and  also  because  the  teeth  of  the  pinion  come  into  contact 
more  frequently,  and  hence  suffer  greater  wear.  In  such  com- 
binations, the  metal  gear  frequently  has  teeth  of  thickness 

less  than  -  and  the  wooden  gear  teeth  of  thickness  greater  than 

-,    to  equalize  strength.      See  Fig.  150.     In  recent  years  gears 

made  of  rawhide  have  been  much  used  for  high  speeds. 
The  blanks  for  rawhide  gears  are  made  by  cementing  specially 
prepared  rawhide  discs  together  under  great  pressure.  Me- 
tallic discs,  on  each  side  of  the  blank,  held  together  by  rivets 
passing  through  the  blank,  assist  the  rawhide  teeth  in  retaining 
their  form.  The  teeth  are  cut  in  the  blank  in  the  same  manner 
that  metallic  teeth  are  cut.  In  using  rawhide  gearing  the 
pinion  is  almost  always  made  of  rawhide  and  the  larger  gear  of 
cast  iron  or  brass.  Such  a  combination  may  be  run  at  a  very  high 
rate  of  speed,  3,000  feet  per  minute  being  a  not  unusual  velocity. 
Rawhide  gears  are  almost  noiseless  in  operation  but  care  must  be 
used  that  they  are  not  subjected  to  extreme  moisture  nor  run  in 
too  dry  an  atmosphere. 

Formerly  it  was  cheaper  to  cast  gear  teeth,  but  the  development 
of  gear-cutting  machinery  has  changed  the  situation  where  a  large 
number  of  gears  with  small  teeth  are  to  be  made.  Modern 
methods  of  gear-cutting  produce  teeth  of  great  accuracy,  and 
have  also  so  greatly  reduced  the  cost  of  production  that  for  high 
speeds,  and  where  smoothness  of  action  is  necessary,  cut  gears 
have  largely  superseded  cast  gears  even  in  large  work. 

There  are  many  methods  of  cutting  gear  teeth  in  practical 
operation,  the  most  common  method  of  cutting  spur  gears  being 
by  the  use  of  a  rotating  cutter.*  The  outlines  of  gear  teeth  vary 
with  the  number  of  teeth  in  the  gear,  the  pitch  or  thickness  of 
tooth  remaining  constant,  and,  theoretically,  a  different  cutter  is 
required  for  every  different  diameter  of  gear  in  a  series  of  the 
same  pitch.  To  meet  this  requirement  would  lead  to  an  excessive 
number  of  cutters  for  each  pitch.  It  is  found  in  practice,  how- 
ever, that  the  same  cutter  can  be  used,  without  serious  error,  for 

*  See  "  Gear-Cutting  Machinery,"  by  Ralph  E.  Flanders. 


TOOTHED   GEARING 


373 


several  sizes  of  gears  of  a  given  pitch.  In  the  system  adopted  by 
the  Brown  &  Sharpe  Mfg.  Co.,  only  24  cutters  are  used  for  each 
pitch  in  the  cycloidal  system,  and  only  8  cutters  for  each  pitch  in 
the  involute  system,  as  given  below.  The  letters  and  numbers 
in  the  first  column  are  the  manufacturer's  designations,  for  pur- 
poses of  ordering  cutters. 


TABLE   XXVII 

CUTTERS  FOR  CYCLOIDAL  TEETH 


Cutter  A    cuts  12  teeth. 


B 

J3 

C 

14 

D 

15 

E 

16 

F 

T7 

G 

18 

H 

19 

I 

20 

J 

21    tO 

22  teeth. 

K 

23     " 

24      " 

L 

2S     " 

26      " 

Cutter  M  cuts     27      to     29     teeth. 

N 

3°            33 

0 

34 

37 

P 

38 

42 

Q 

43 

49 

R 

5° 

59 

S 

60 

74 

T 

75 

99 

U 

100 

149 

V 

150 

249 

w 

2=50    or  more 

X 

Rack. 

TABLE   XXVIII 

CUTTERS  FOR  INVOLUTE  TEETH 

Cutter  No.  i  cuts  from  134  teeth  to  rack. 


2 

3 
4 
5 

55  tc 
35  ' 
26  ' 

21  ' 

134  tee 

54 
34 
25 

20 

16 

th. 

6 

7 

Q 

J7  ' 

14  ; 

When  gear-cutting  is  carefully  done,  very  accurate  work  may 
be  accomplished.  It  is  to  be  noted,  however,  that  the  form  of 
the  teeth  when  cut  with  a  set  of  cutters,  as  above,  are  not  all 
theoretically  correct;*  and  even  in  best  practice  the  error  in  the 
gear-cutting  machine  itself,  coupled  with  that  due  to  dullness  of 
cutters  and  deviation  due  to  different  degrees  of  hardness  in  the 
metal,  may  be  considerable. 

172.  Forces  Acting  on  Spur  Gears.  In  Fig.  151  let  the  gear  A 
drive  the  gear  B.  Let  Fa  be  the  velocity  of  the  pitch  circle  of  A; 
and  Fb  be  the  velocity  of  the  pitch  circle  of  B.  Also  let  W&  be 
the  equivalent  driving  force  acting  at  the  pitch  circle  of  A,  and 
let  Wb  be  the  equivalent  resisting  force  acting  at  the  pitch  circle 

*  There  are  gear-cutting  machines  which,  theoretically,  generate  correct  forms 
of  teeth  for  all  gears  of  a  series. 


374 


MACHINE   DESIGN 


of  B.  If  now  the  tooth  outlines  are  properly  constructed,  the  line 
of  action  of  the  actual  driving  force  W^  will  always  pass  through 
the  pitch  point  and  the  angular  velocity  ratio  of  A  to  B  will  be 
constant.  The  action  of  the  pitch  circles  will  be  as  though  they 
rolled  upon  each  other  and  their  linear  velocity  will  be  the  same 
or  Fa  =  Fb.  From  the  principle  of  work 

w*  v*  =  W*  Fb     Therefore  WA  =  Wb 

The  tangential  driving  force  exerted  by  one  gear  upon  another 
is,  therefore,  independent  of  the  angle  of  pressure,  in  any  correct 


FIG.  i;i. 


FIG.  153. 


system  of  gearing,  and  the  action  is,  in  this  respect,  the  same  as 
if  a  pair  of  teeth  were  continually  in  action  at  the  pitch  point. 

The  distribution  of  the  reaction  at  the  bearings  due  to  the 
pressure  between  teeth  (W19  Fig.  151),  and  its  bending  effect  on 
the  shaft  which  supports  B,  will  depend  upon  the  relative  positions 
of  the  gear  and  bearings;  but  the  latter  will,  in  any  case,  be 
directly  proportional  to  W ,.  As  the  obliquity  of  the  line  of 
action  C  D  is  increased,  the  angle  B  (Fig.  151)  is  increased  and 
hence  sec  6  is  also  increased.  Therefore,  since  Wl  =  TFa  sec  0, 


TOOTHED   GEARING  375 

the  pressure  on  the  bearings  is  increased  with  an  increase  on  the 
obliquity  of  the  line  of  action;  but  the  torque  on  the  driven  shaft 
remains  unchanged. 

In  cycloidal  gearing  the  obliquity  varies  from  a  maximum  at 
the  beginning  of  the  contact  to  zero  when  the  contact  point  lies 
in  the  line  of  centres;  and,  during  the  arc  of  recess,  it  increases 
to  a  maximum  at  the  end  of  contact.  The  maximum  value  of 
the  angle  6,  Fig.  148,  is  about  22°,  with  usual  forms  of  cycloidal 
teeth.  When  0  equals  22°,  sec  0  equals  1.08,  or  the  maximum 
normal  pressure  is  about  8  per  cent  greater  than  the  tangential 
rotative  force. 

The  obliquity  is  constant  throughout  the  arc  of  action  in 
involute  gears,  and  the  angle  0,  Fig.  147,  is  usually  14^2°  °r  I5°- 
When  0  =  15°,  sec  0  =  1.035,  or  the  normal  pressure  is  3^  per 
cent  greater  than  the  tangential  force.  In  the  above  discussion 
the  influence  of  friction  has  been  neglected.  During  the  arc  of 
approach  the  frictional  force  F  (Fig.  151)  deflects  the  line  of 
action  of  W \  in  such  a  way  as  to  increase  the  effective  obliquity. 
During  the  arc  of  recess  it  acts  in  the  opposite  direction  and  de- 
creases the  obliquity.  The  influence  of  this  frictional  force  is 
small  and  may,  usually,  be  neglected,  but  its  action  accounts, 
to  a  certain  degree,  for  the  well-known  fact  that  gears  run 
more  smoothly  during  recess  than  during  approach. 

It  is  usually  intended  that  more  than  one  pair  of  teeth  shall 
be  in  action  at  all  times,  but,  owing  to  the  unavoidable  inaccuracy 
of  form  and  spacing  previously  noted,  it  is  not  safe  to  depend  upon 
a  distribution  of  the  load  between  two  or  more  teeth  of  a  gear.  It 
is  safest  to  provide  sufficient  strength  for  carrying  the  entire  load 
on  a  single  tooth.  In  the  rougher  classes  of  work,  this  load  may 
be  concentrated  at  one  end  of  the  tooth,  as  indicated  in  Fig.  152, 
and  all  such  gears  should  be  carefully  inspected  and  corrected, 
if  intended  to  carry  heavy  and  important  loads.  With  well 
supported  bearings,  and  machine-moulded  or  cut  gears,  it  is  not 
unreasonable  to  consider  the  load  as  fairly  well  distributed  across 
the  face  of  the  gear,  if  the  face  does  not  exceed  in  width  about 
three  times  the  circular  pitch  (see  Fig.  153). 

The  obliquity  of  the  line  of  pressure  gives  rise  to  a  crushing 


376 


MACHINE  DESIGN 


action  on  the  teeth  (due  to  the  radial  component  of  the  normal 
force),  in  addition  to  the  flexural  stress  which  results  from  the 
tangential  component.  This  crushing  component,  with  the 
ordinary  proportions  of  teeth,  does  not  exceed  10  per  cent  of  the 
normal  pressure.  Its  effect  is  to  reduce  the  tensile  stress  due  to 
flexure,  and  to  increase  the  compressive  stress.  Since  cast  iron 
is  far  stronger  in  compression  than  in  tension,  this  may  be 
neglected  in  gears  made  of  that  metal,  while  in  the  case  of  steel, 
or  composition  gears,  the  margin  of  safety  assumed  usually  makes 
it  unnecessary  to  consider  this  component. 

173.  Strength  of  Spur  Gear  Teeth.  The  assumption  often 
made  that  the  teeth  of  spur  gears  can  be  considered  as  rectangular 
cantilevers,  in  determining  their  strength,  is  not  satisfactory,  es- 
pecially when  treating  pinions  having  a  small  number  of  teeth.  Fig. 
154  shows  four  gear  teeth  which  have  the  same  thickness  at  the 


(a) 


(b) 


(d) 


FIG.  154 


pitch  line  and  the  same  height.  The  tooth  marked  (a)  is  one  of 
an  involute  rack;  (b)  is  one  of  an  involute  pinion  having  12 
teeth;*  (c)  is  one  of  a  cycloidal  gear  having  30  teeth;  (d)  is  one 
of  a  cycloidal  pinion  of  12  teeth. 

Mr.  Wilfred  Lewis,  of  Wm.  Sellers  &  Co.,  seems  to  have  been 
the  first  to  investigate  the  strength  of  gear  teeth  with  due  regard 
to  the  actual  forms  used  in  modern  gearing.  His  work  was 
published  originally  in  the  Proceedings  of  the  Engineers'  Club  of 
Philadelphia,  in  January,  1893,  and  his  method  of  investigation 
was  as  follows :  Accurate  drawings  of  gear  teeth  were  made  on  a 
large  scale,  and  the  line  of  action  of  the  normal  force,  when  acting 
on  the  point  of  a  tooth  was  drawn  in;  see  Fig.  155.  From  the 

*The  i2-tooth  involute  pinion  may  have  its  teeth  weakened  by  a  correction 
for  interference;  but  it  is  usually  better  to  correct  the  points  of  the  mating  gear. 


TOOTHED   GEARING 


377 


intersection  of  this  line  of  action  with  the  centre  line  of  the  tooth, 
a  parabola  was  drawn  tangent  to  the  sides  of  the  tooth,  thus 
locating  a  beam  of  uniform  strength  equal  to  the  effective  strength 
of  the  tooth  (see  Article  15).  The  points  of  tangency  a,  a,  locate 
the  weakest  section  of  the  tooth,  and  the  bending  moment  applied 
to  this  section  is  W  I.  Then  from  equation  J,  page  94. 

^  /  _  pb(2hY  _  2 

'  e  6  ^ 


Wl 


=  bps(y) 


(i) 


Where  b  =  the  breadth  of  the  tooth  in  inches,  p  =  the  tensile 

stress,  and  s  =  the  circular  pitch.     The  factor  y  is  a  variable, 

depending  on  the  shape  of  the  tooth.      Mr. 

Lewis    found    that    its   value    is    practically 

independent  of  the   pitch    (since   s,  h  and  I 

are   proportional   to   the  pitch),  but  depend- 

ent  mainly  on  the  number  of  teeth  in   the 

gear.    Tabulated  values  of  this  coefficient  may 

be  found  in  Kent's  "  Mechanical  Engineers' 

Pocketbook,"  page   901.     From   these    tabu- 

lated values,  Mr.  Lewis  deduced  the  follow- 

ing equations  in  which  N  =  the  number  of  teeth  in  the  gear. 

For  the  15°  involute  system  and  the  cycloidal  system  using 
a  generating  circle  whose  diameter  equals  the  radius  of  the  12- 
tooth  pinion, 


FIG.  i^ 


For  the  20°  involute  system, 


•      (3) 


Mr.  Lewis'  investigations  on  cycloidal  gears  were  made  on  a 
system  using  the  radius  of  the  12 -tooth  pinion  as  the  diameter 
of  the  describing  circle.  Modern  practice  sometimes  makes  the 
radius  of  the  i5-tooth  pinion  the  diameter  of  the  describing  circle, 
which  gives  somewhat  weaker  teeth  than  the  first  system.  The 


378  MACHINE   DESIGN 

difference  is  small,  however,  compared  to  the  variation  in  the 
assumed  stress,  p,  and  since  cycloidal  teeth  are  now  little  used 
for  small  and  moderate-sized  gears,  equation  (2)  will  be  adopted 
in  this  work  for  standard  gears. 

The  Lewis'  formula  is  convenient  for  determining  W,  b,  s,  or  />, 
where  the  number  of  teeth  (N)  is  known;  but  a  very  common 
problem  in  design  is  to  determine  the  pitch  (s),  when  the  pitch 
diameter  of  the  gear  is  given  and  the  number  of  the  teeth  is  un- 
known. The  formula  may  be  adapted  to  this  last  stated  problem 
as  follows.*  To  accord  with  modern  practice,  circular  pitch  will 
also  be  transformed  to  diametral  pitch. 

Let  D  =  the  pitch  diameter. 
"     w  =  the  load  per  inch  of  face. 

"     S  =  the  diametral  pitch  =  —  or  s  =  -7, 

s  S 

Then  N  =  D  X  S 


Therefore  W  =  bsp  (.124  -  '^)  =  b  X  ^  X  p  (.124-  ^ 


or  since  w  =  W  +  b 


*-,(*- S3 


and  therefore  S  =  -^-(.194  +  J   0.8  _  2'T5W)      ...     (6) 
w  \  v  ' 


w  \  v  p  D 

The  pitch  can  be  found  from  equation  (6)  for  any  values  of  w, 
D,  and  p,  when  the  face  of  the  gear  is  known  or  assumed.  A 
common  problem  is  as  follows:  The  distance  between  two 
shafts  and  their  velocity  ratio  is  known;  required  the  pitch  of  spur 
gears  to  connect  these  shafts  for  a  given  load  and  working  stress 
on  the  teeth.  The  centre  distance  of  the  shafts,  and  the  velocity 
ratio  fix  the  diameter  of  the  gears.  The  face  of  the  gears  may 
be  governed  by  the  space  available,  or  it  may  be  assumed  by  the 
designer  upon  other  considerations.  To  illustrate;  suppose 

*  See  a  discussion  by  John  H.  Barr,  Trans.  A.  S.  M.  E.,  Vol.  XVIII,  page  766. 


TOOTHED    GEARING  379 

W  =  15,000  Ibs.,  p  =  8,000  Ibs.  per  square  inch,  and  that  the 
smaller  gear  is  to  be  40  inches  diameter.  Assume  also  that  the 
face  of  the  gear  may  be  taken  as  6  inches.  The  load  per  inch  of 
face  is  w  =  15,000  -r-  6  =  2,500  Ibs.,  hence, 


500  ,000x40 


=    i.i    or    say    i 


diametral  pitch. 

The  diagrams  shown  in  Figs.  156  and  157  are  plotted  from 
equation  (5).  That  in  Fig.  .156  covers  the  range  from  12  to  6 
diametral  pitch  and  Fig.  157  covers  the  range  from  5  to  i  diametral 
pitch.  The  abscissas  (Scale  A)  represent  pitch  diameters  of  gears 
in  inches,  and  the  ordinates  (Scale  B)  the  load  in  pounds  per  inch 
of  width  of  face,  for  a  stress  0/6,000  pounds  per  square  inch.  Any 
other  stress  could  have  been  taken  for  plotting  the  diagrams,  and 
any  other  may  be  used  in  solving  problems  by  them.  A  curve 
is  drawn  for  each  pitch;  to  illustrate,  let  S  =  1.5,  let  p  =  6,000. 
Substituting  these  values  in  (5) 

2.  ic 


or 


/.xog             2.  is      \ 
w  =  6,000  (--      - 2—  ) 

V   1.5  D   X    2.2$) 

w  =  (^SS6 Jjf-) 


hence  when  Z>  =  3.7,  w  =  o;      D  =  10,  w  =  983;      D  =  20, 
w  =  1,270,  etc. 

Plotting  these  corresponding  values  of  D  and  w  as  abscissas 
and  ordinates,  respectively,  the  curve  for  a  diametral  pitch  =  i# 
is  drawn  through  the  points.  The  other  curves  are  constructed 
in  a  similar  manner.  If,  then,  the  diameter  of  the  gear  is  known, 
the  allowable  load  per  inch  of  face  for  a  stress  of  6,000  pounds 
per  square  inch  may  be  found  bypassing  vertically  upward  from 
the  given  diameter  on  scale  A,  to  the  curve  corresponding  to  the 
pitch,  and  then  moving  horizontally  to  the  left-hand  scale  B, 
which  gives  the  required  load  per  inch  of  face.  Scale  B  is  re- 
produced at  thq  top  of  the  diagram,  as  scale  C,  and  a  45°  diagonal 
marked  6,000  is  drawn  from  the  lower  right-hand  corner  of  the 


MACHINE   DESIGN 


diagram  to  scale  C.     If,  then,  instead  of  moving  horizontally 
from  the  pitch  curve  to  scale  B  on  the  left,  the  movement  be 


Scale  B 


horizontally  to  the  right  (or  left)  to  the  diagonal  marked  6,000, 
and  then  vertically  upward  to  the  scale  C,  the  same  reading  will 
be  obtained  on  C  as  originally  found  on  B. 


TOOTHED  GEARING 


Furthermore,  if  other  diagonals  be  drawn,  as  shown,  from 
various  points  on  scale  C,  they  may  be  used  to  read  loads  per  inch 


Scale  B 


of  face  for  stresses  corresponding  to  these  respective  points  on 
this  scale,  since  from  equation  (4)  it  appears  that  the  stress 
varies  directly  with  the  load.  These  stresses  are  indicated  along 


382  MACHINE   DESIGN 

the  several  diagonals.  Thus  to  find  the  pitch  when  w  =  2,500, 
p  =  8,000,  D  =  40,  from  2,500  on  scale  (C),  Fig.  157,  pass  verti- 
cally downward  to  the  diagonal  marked  p  =  8,000;  then  horizon- 
tally to  a  point  on  the  vertical  rising  from  D  =  40",  on  scale  A. 
The  pitch  curve  nearest  this  point  is  for  i  diametral  pitch  which 
would  be  the  required  pitch. 

If  it  is  required  to  find  the  load  per  inch  of  face  for  a  gear  of 
given  diameter  and  pitch,  with  an  assigned  stress,  start  at  the 
point  on  scale  (A),  corresponding  to  the  diameter;  pass  upward  to 
the  given  pitch  curve;  thence  horizontally  (right  or  left  as  the  case 
may  be),  to  the  proper  stress  diagonal;  thence  upward  to  scale 
(C),  where  the  unit  load  may  be  read  off.  (See  Fig.  156  or  157.) 

If  the  diameter,  pitch,  and  unit  load  are  known  quantities, 
pass  upward  from  the  diameter  reading  on  scale  A,  to  the  proper 
pitch  curve;  thence  horizontally  to  a  point  under  the  unit  load 
on  scale  C,  and  the  stress  may  be  found  by  interpolating  between 
the  adjoining  diagonals. 

It  may  be  noticed  that  the  different  pitch  curves  in  either  Fig. 
156  or  157  have  a  common  tangent  through  the  Origin  O.  The 
points  of  tangency  correspond  to  the  diameters  of  gear  at  which 
cycloidal  teeth  have  radial  flanks  for  their  respective  pitches;  i.e., 
a  i2-tooth  gear.  The  various  curves  have  not  been  extended  far 
beyond  this  point  as  in  that  case  they  intersect,  and  mar  the 
clearness  of  the  diagram.  Since  this  intersection  occurs  after 
the  diameter  of  the  1 2-tooth  gear  is  reached,  it  is  evident  that  the 
remainder  of  the  curve  is  of  no  practical  importance.  In  fact, 
increase  of  pitch,  or  a  less  number  of  teeth  for  a  given  diameter, 
beyond  this  point  will  not  give  additional  strength,  because  the 
length  of  tooth  will  be  increased,  and  the  flanks  will  be  under- 
cut to  an  extent  which  more  than  compensates  for  the  added 
thickness  of  tooth. 

The  data  may  be  such  that  a  point  corresponding  to  a,  Fig. 
157,  will  lie  to  the  left  and  above  all  the  pitch  curves,  i.e.,  above 
the  common  tangent  through  O.  If  the  same  data  were  substituted 
in  equation  (4)  an  imaginary  quantity  would  result.  This  means 
that  the  unit  load  taken  cannot  be  carried  with  the  stress  and 
diameter  assumed,  by  any  possible  pitch.  The  only  recourse  for 


TOOTHED   GEARING 


383 


a  gear,  of  the  given  diameter  and  total  load  TF,  is  to  increase  the 
face,  and  thus  reduce  w,  or  to  use  a  material  which  permits  a 
higher  intensity  of  stress. 

It  should  be  noted  that  the  teeth  of  the  smaller  gear  of  a  mating 
pair  are  weaker  in  form  than  those  of  the  larger.  The  wear,  also, 
is  greater  on  the  teeth  of  the  smaller  gear  since  they  come  in  con- 
tact more  frequently.  Hence,  in  general,  if  the  small  gear  is 
properly  designed  the  larger  gear  will  have  sufficient  strength. 


FIG.  158. 

This  does  not  apply  to  certain  forms  of  reinforced  or  "shrouded" 
teeth  discussed  later,  nor,  necessarily,  where  the  thickness  of 
teeth  and  spaces  are  unequal,  nor  where  the  mating  gears  are  of 
different  material. 

174.  Strength  of  Bevel  Gear  Teeth.  If  a  pair  of  bevel  gear 
teeth,  Fig.  158,  have  just  come  into  contact  as  shown  at  a,  Fig. 
147,  then  the  driving  force  is  applied  to  the  point  of  the  driven 
tooth  by  the  root  of  the  driver.  The  tooth  of  the  driven  wheel 
will  be  deflected  a  certain  amount,  while  the  deflection  of  the  driv- 
ing tooth  will  be  negligible.  Since  the  deflection  of  the  driven  tooth 
is  caused  by  the  rotative  effort  of  the  driving  gear,  the  magnitude 


384  MACHINE   DESIGN 

of  this  deflection  at  any  point  on  the  line  of  contact  of  the  two 
teeth  will  be  proportional  to  the  movement  of  the  corresponding 
contact  point  of  the  driver  or  to  its  distance  from  the  axis  of 
rotation  of  the  driver;  hence,  from  similar  triangles,  will  also  be 
proportional  to  the  distance  from  its  own  axis  of  rotation.  Now 
the  cross-sectional  outlines  of  the  tooth  are  similar  at  all  points, 
and  it  can  be  shown  that  in  the  case  of  simple  cantilevers  of 
similar  form  the  load  applied  is  proportional  to  deflection.  It 
has  just  been  shown,  however,  that  the  deflection  of  the  tooth  at 
any  point  is  proportional  to  the  distance  of  that  point  from  the 
axis  of  rotation.  Hence  the  load  on  the  tooth  at  any  point  must 
also  be  proportional  to  the  distance  from  the  axis,  being  least  at 
the  small  end,  and  greatest  at  the  large  end,  the  mean  value  be- 
ing at  the  middle  of  the  tooth.  Therefore  a  spur  gear  which  has 
the  same  width  of  face,  and  teeth  of  the  same  form  and  pitch 
as  the  mean  section,  will  have,  theoretically,  the  same  strength  as 
the  bevel  gear.  It  can  also  be  shown  that  in  simple  cantilevers 
of  equal  breadth  and  similar  outline,  the  stresses  induced  at  cor- 
responding points  on  the  cantilevers  are  equal,  if  the  load 
applied  is  proportional  to  the  linear  dimensions.  Hence  the 
maximum  stresses  are  the  same  at  all  sections  of  the  tooth. 

It  is  evident  that  the  relation  established  above  between  the 
mean  section  of  a  bevel  gear  and  a  spur  gear  with  similar  teeth, 
may  be  (and  often  is)  used  as  a  means  of  designing  bevel  gears. 
It  is  much  more  convenient,  however,  to  deal  with  the  teeth  at 
the  outer  or  large  end.  If,  also,  the  pitch  radii  are  used  instead 
of  the  addendum  radii,  the  error  will  not  be  great. 

Let  r:  =  the  pitch  radius  at  the  small  end  of  the  tooth. 

"  r2  =  the  pitch  radius  at  the  large  end  of  the  tooth. 

"  r    =  the  mean  pitch  radius  of  the  tooth. 

"  b   =  the  width  of  the  face  of  the  tooth  along  the  elements. 

"  w  =  the  load  per  inch  of  face  at  the  radius  r. 

"  w2=  the  load  per  inch  of  face  at  the  radius  r2. 

"  W=  the  resultant  load  on  the  tooth  =  w  b. 

11  we=  the  equivalent  load  per  inch  of  face  which,  if  acting 
at  a  radius  r2,  would  produce  the  same  rotative  effect  as  the 
actual  load. 


TOOTHED  GEARING  385 
Since  the  load  on  the  tooth  varies  as  the  radii,  the  total  re- 
sultant load  will  act  at  a  radius  R  =  -     22 \  ,  and  the  torsional 

moment  due  to  the  resultant  force,  W,  will  be 


3          <2  —  *\) 
Now  by  definition  wv  b  r  2  =  W  R. 

Therefore,  v.-'-v&L     ....      .;.:..     (7) 


Also,  since  the  load  varies  with  the  radius, 

W.£          W  W  2W  2Wr2 


' 


2 
And  from  (7)  and  (8) 

g  =  jyji  _  iwJ^r-!^L  _  3  fr-_^*  -  k    (9) 

The  actual  load,  w2,  will  always  be  greater  than  w  in  the  ratio 
shown  above.  A  bevel  gear,  therefore,  will  be  more  heavily 
loaded  at  the  large  end  than  a  spur  gear  of  the  same  diameter,  and 
carrying  the  same  torque,  in  the  ratio  shown  above.  If,  however, 
we  is  known,  w2  can  be  computed,  and  used  in  equations  (5)  and 
(6)  of  Art.  173  instead  of  w.  Usually  r±  is  made  not  less  than 

2  2  W^ 

—r2.   When  rt  =  —r2,   -  -  =  k  =  1.4,  and  this  value  can  be  used 

o  O  e 

in  computing  w2  unless  the  face  of  the  gear  is  excessively  long. 
It  should  be  especially  noted  that  in  solving  problems  in  bevel 
gearing,  either  by  the  diagrams  of  Figs.  156  and  157  or  by 
equations  (5)  and  (6)  of  Art.  173,  the  diameter  D,  which  must 
be  substituted  therein,  is  that  corresponding  to  the  formative 
circle,  whose  radius  is  R<  =  r2  sec  0,  Fig.  158,  as  the  form  of 
the  tooth  is  fixed  by  this  radius  and  not  by  the  radius  r2.  The 

*  See  also  Mr.  Lewis's  article,  Proceedings  Engineers'  Club  of  Philadelphia, 
Jan.,  1893. 
25 


386  MACHINE  DESIGN 

computations  should  be  made  for  the  smaller  of  the  two  gears,  as 
in  the  case  of  spur  gears. 

Example.  Design  a  pair  of  bevel  gears  to  transmit  50  H.-P. 
with  a  velocity  ratio  of  3  to  2;  the  gears  to  be  of  cast  iron,  and 
the  maximum  fibre  stress  to  be  4,000  pounds  per  square  inch. 
The  revolutions  per  minute  of  the  shafts  are  to  be  300  and  200 
respectively. 

Lay  off  the  axes  OV  and  OT,  Fig.  158,  and  draw  OU  so 
that  corresponding  radii  of  N  and  M  are  in  the  proportion  of  3 
to  2.  Then  it  is  found  that  0  =  34°  and  0'  =  56°.  Assume  tenta- 
tively that  r2  for  the  large  gear  =  15"  and  r2  for  the  small  gear 
=  10",  and  take  a  trial  width  of  face  of  4". 

2    X   7T    X    10   X   300 

The  velocity  at  the  radius  r2  =  —  ~=I>575 

feet  per  minute.     Hence  the  equivalent  total  load  at  a  radius  r2 


4 

Therefore  w2  =  we  X  k  =  262  X  1.4  =  367  pounds  per  inch  of 
face.  Since  sec  0  =  sec  34°  =  1.2,  the  diameter  of  the  formative 
circle  is  20  X  1.2  =  24"  and  from  the  diagram,  Fig.  157,  or  from 
equation  (6),  it  is  found  that  for  D  =  24",  p  =  4,000  pounds, 
and  w  =  367  pounds,  the  diametral  pitch  is  very  nearly  3^, 
which  may  therefore  be  selected.  This  would  give  70  teeth  for 
the  small  gear  and  105  for  the  large  gear.  The  width  of  face  is  a 
little  more  than  three  times  the  circular  pitch  and  is  therefore  in 
accordance  with  good  practice. 

175.  Stresses  in  Gear  Teeth.  Gear  teeth  of  all  kinds  are  likely 
to  be  subjected  to  shock,  unless  running  at  a  very  low  velocity, 
and  the  danger  from  shock  increases  as  the  velocity  increases; 
hence  the  allowable  stress  must  be  reduced  as  the  velocity  is 
increased.  Reliable  experimental  data  on  the  allowable  stress 
in  gear  teeth  are  lacking,  although  many  empirical  rules  are  to 
be  found  in  treatises  on  the  subject.  The  values  given  by  Mr. 
Lewis,  in  the  paper  already  quoted,  are  probably  as  reliable  as 
any,  for  teeth  that  bear  along  their  entire  length.  The  following 
equations  have  been  deduced  from  Lewis's  work: 


TOOTHED  GEARING  387 

For  cast  iron,  p  =  8,000  (-  -  -\     .      .     (10) 


for  steel  p  -  20,000  •     •     (") 


where  V  =  the  velocity  at  pitch  line  in  feet  per  minute.  It  would 
be  probably  safe  to  take  the  stress 

/      600     \ 
for  bronze  as  p  =  12,000   [-  -  -  1      .      .      (12) 

since  the  resilience  of  bronze  is  greater  than  that  of  cast  iron. 
An  old  empirical  rule  for  rough  cast  teeth  is 

W  =  200  X  s  X  b     .      .     ...      .      (13) 

where  W,  as  before,  is  the  total  load,  s  the  circular  pitch,  and  b 
the  width  of  face  of  tooth.  The  strength  of  wooden  mortise  teeth, 
made  of  beech  or  maple,  may  be  taken  as  about  one-half  that  of 
cast  iron,  under  the  same  circumstances;  and  the  strength  of 
good  rawhide  gears  may  be  taken  as  equal  to  that  of  similar  gears 
made  of  cast  iron.  It  is  to  be  noted  that  a  rawhide  gear  will  en- 
dure considerable  more  shock  than  one  made  of  cast  iron. 

While  rough  cast  teeth  are  more  likely  to  bear  on  one  corner 
only,  they  are  stronger  than  cut  teeth  of  the  same  pitch, 
which  compensates  in  a  measure  for  this  defect;  furthermore, 
there  is,  usually,  an  excess  of  strength,  to  allow  for  wear,  in  all 
new  gears,  and  the  subsequent  wear  tends  to  correct  the  initial 
unequal  bearing,  along  the  elements. 

On  account  of  the  increased  liability  to  shock,  with  increase 
of  speed,  and  also  because  of  the  noise  of  operation  at  high  speeds, 
there  is  a  limit  to  the  speed  at  which  any  form  of  gear  may  be 
safely  and  conveniently  operated.  Mr.  A.  Fowler,  in  Engineering, 
April,  1889,  gives  the  following  as  maximum  values  at  which 
gearing  may  be  successfully  operated: 

Ft.  per  Min. 

Ordinary  cast-iron  gears  ...................................  1,800 

Helical  cast-iron  gears    ....................................  2,400 

Mortise  wheel  and  cast-iron  pinion    .........................  2,400 

Ordinary  cast-steel  gears    ..................................  2,400 

Helical  cast-steel  gears  .....................................  3,ooo 

Special  cast-iron  machine-cut  gears  ..........  .  .  ............  3,ooo 


388  MACHINE   DESIGN 

Although  higher  velocities  are  occasionally  found  in  practice, 
these  are  undoubtedly  maximum  average  values  and,  in  general, 
the  velocity  should  not  be  more  than  two-thirds  the  values  given 
above,  on  account  of  noise  and  wear.  Rawhide  gearing,  which 
operates  almost  noiselessly,  may  be  run  satisfactorily  up  to  3,000 
feet  per  minute. 

176.  Width  of  Face  of  Gears.  Equation  (5)  of  Art.  173  gives 
the  load  per  inch  of  face  that  may  be  applied  to  a  tooth  of  the 
given  form  and  pitch,  the  total  load  depending  on  the  width  of 
face  as  shown  by  equation  ^(4)  of  the  same  article.  The  dur- 
ability of  a  tooth  for  a  given  load  is,  therefore,  theoretically 
increased  with  increase  of  face.  The  difficulty  of  securing 
uniform  distribution  of  the  load  along  the  contact  element  in- 
creases, however,  as  the  width  of  face  is  increased,  and  this  imposes 
a  practical  limit  to  the  width  of  the  face.  On  the  other  hand,  if  the 
intensity  of  the  load  on  the  tooth  is  too  great,  excessive  wear 
may  result.  The  equations  given  above  do  not  take  wear  into 
account,  the  allowable  load  being  fixed  with  reference  to  the 
stress  alone.  On  this  basis  a  large  tooth  may  carry  a  much 
higher  load  per  inch  of  face,  but  the  wear  will  be  proportion- 
ally greater,  the  velocity  being  the  same.  The  empirical  rule 
given  in  equation  (13)  of  Art.  175  assigns  a  load  of  200  pounds 
per  inch  of  face,  per  inch  of  circular  pitch.  For  a  tooth  of  i" 
circular  pitch  this  load  will  give,  by  the  Lewis  equation,  a  stress 
of  only  2,000  pounds  per  square  inch,  for  moderate -sized  gears. 
This  is  a  very  low  stress,  for  ordinary  speeds,  so  that  this  rule 
would  give  more  durable  teeth  than  the  Lewis  equation,  as 
ordinarily  applied. 

Experimental  data  on  the  durability  of  teeth  are  lacking.  It 
is  evident,  however,  that  the  allowable  load  will  depend  largely 
on  the  character  of  the  service,  velocity  of  rubbing,  lubrication, 
and  the  material  used.  Thus,  for  ordinary  cut  cast-iron  teeth 
under  constant  service,  the  value  given  above  (200  Ibs.)  is  prob- 
ably conservative;  while  with  teeth  of  high-grade  steel  much 
greater  loads  may  be  carried.  Cases  are  on  record  where  loads 
of  over  2,000  pounds  per  inch  of  face  were  successfully  carried, 
with  a  peripheral  velocity  of  over  2,000  feet  per  minute,  the 


TOOTHED   GEARING  389 

pinion  being  of  forged  steel  and  the  gear  a  steel  casting,  4.92" 
circular  pitch.  Well-made  gears  of  rawhide  may  be  loaded  up 
to  150  pounds  per  inch  of  face,  per  inch  of  circular  pitch;  but  in 
no  case  should  the  load  exceed  250  pounds  per  inch  of  face.* 
In  the  case  of  machines  such  as  punching-machines  which 
work  intermittently,  and  whose  operation  extends  over  a  short 
space  of  time,  the  element  of  wear  is  not  so  important  in  the  de- 
sign of  the  teeth;  but  in  such  gears  as  those  connecting  street- 
railway  or  automobile  motors  with  the  driving  axles,  where  the 
work  is  both  continuous  and  severe,  wearing  qualities  may  be 
fully  as  important  as  strength;  and  gears  made  of  steel  or  other 
hard  materials  may  have  to  be  used  solely  on  this  account. 

Good  practice  makes  the  face  of  the  tooth  about  three  times 
the  circular  pitch ;  but  in  fixing  the  pitch  and  width  of  face,  in 
extreme  cases,  the  points  discussed  above  should  be  con- 
sidered. 

177.  Other  Forms  of  Gear  Teeth.  Gear  teeth  made  according 
to  the  Brown  &  Sharpe  standard,  on  which  the  foregoing  dis- 
cussion is  based,  have  been  found  very  satisfactory  for  average 
conditions,  and  are  in  most  common  use  in  this  country.  For 
extreme  conditions,  however,  it  has  been  found  necessary  to 
reinforce  such  teeth,  or  to  use  teeth  of  a  different  form. 

A  very  common  way  of  reinforcing  teeth  of  cast  gears  is  by 
shrouding,  which  consists  in  casting  an  annular  ring  of  metal  on 
one  or  both  ends  of  the  teeth,  as  shown  in  Fig.  159.  This  ring  is 
cast  as  an  integral  part  of  the  gear  casting,  and  hence  strengthens 
the  gear  tooth  by  practically  twice  the  shearing  strength  of  the 
cross-section  of  the  tooth,  when  both  ends  are  shrouded  to  the 
top.  The  teeth  of  the  pinion  are,  from  their  outline,  always 
weaker  than  those  of  the  gear,  and  the  wear  on  them  is  also 
greatest.  The  shrouding  should,  therefore,  be  put  on  the  pinion; 
and  if  carried  to  the  top  of  the  tooth  on  both  ends  it  will  give 
them  an  excess  of  strength  over  those  of  the  gear,  with  usual 
widths  of  face.  If  the  gears  to  be  reinforced  do  not  differ  greatly 
in  diameter,  the  teeth  of  both  may  be  shrouded  half  way  up. 

*  Private  communication  from  the  New  Process  Raw  Hide  Co. 


390 


MACHINE   DESIGN 


Shrouding  is  used  mostly  on  rough  cast  gears,  the  shroud  practi- 
cally prohibiting  the  cutting  of  the  teeth  by  the  usual  methods. 

If  the  gears  are  to  run  in  one  direction  only,  and  where  very 
heavy  pressures  are  to  be  withstood,  a  form  of  tooth  as  shown  in 
Fig.  160,  and  known  as  a  buttress  tooth,  may  be,  but  seldom  is, 
employed.  The  driving  face,  A,  is  made  of  correct  theoretical 
outline,  while  the  back  face  B  may  be  of  any  outline*  that  will 
give  the  required  strength,  and  clear  the  teeth  of  the  mating  gear. 
The  front  face  should  be  of  standard  cycloidal  or  involute  form 
and  the  backs  are  preferably  involute  forms,  with  a  much  greater 
obliquity  of  generator  than  would  be  permissible  in  driving. 

For  some  time  past  there  has  been  a  marked  tendency  f  on 
the  part  of  the  designers  of  gearing  for  extremely  trying  service 
to  depart  from  the  Brown  &  Sharpe  standard,  and  to  use  teeth 
somewhat  shorter  than  those  given  by  that  standard.  In  some 
instances  the  same  angle  of  pressure  has  been  retained,  while  in 
others  this  angle  has  been  increased.  Mr.  C.  W.  Hunt  reported 
to  the  A.  S.  M.  E.  in  1897  (Vol.  XVIII)  the  results  of  the  adop- 
tion of  such  a  system  and  gives  full  information  for  their  design. 
A  few  other  manufacturers  have  adopted  similar  systems.  The 
need  of  a  small  gear  of  great  strength,  in  automobile  work,  has 
increased  the  demand  for  a  stronger  form  of  tooth,  and  it  would 
seem  that  the  old  standard  must  be  modified  or  a  second  standard 
adopted  for  extreme  service.  The  most  prominent  form  of  these 
so-called  "  stub  teeth,"  at  present,  is  that  advocated  by  the 
Fellows  Gear  Shaper  Co.  In  this  system  an  involute  tooth 
with  a  pressure  angle  of  20°  is  used,  the  addendum  being  about 
0.8  as  high  as  that  of  the  Brown  &  Sharpe  standard.  This 
gives  a  tooth  nearly  twice  as  strong  as  the  old  standard.  Some- 
times these  stub  teeth  are  given  the  height  of  a  standard  tooth  of 
smaller  pitch;  thus  a  6-pitch  stub  tooth  may  have  the  length  of 
a  standard  8-pitch  tooth,  in  which  case  the  gear  is  sometimes 
described  as  a  6-8  gear.  Notwithstanding  the  fact  that  the  arc 
of  contact  in  stub-tooth  gears  is,  generally  speaking,  less  than  in 

*  See  "  Kinematics  of  Machinery,"  by  John  H.  Barr,  page  131. 
f  See  a  paper  by  R.  E.  Flanders,  Trans.  A.  S.  M.  E.,  Vol.  XXX,  resume  of 
other  systems. 


TOOTHED   GEARING 


391 


the  old  standard,  they  run  well  and  will  undoubtedly  be  more 
used  in  the  future. 

178.  Strength  of  Gear  Rims  and  Arms.  The  rim  of  the  gear 
wheel  must  not  only  be  strong  enough  to  resist  the  forces  brought 
upon  it,  but  stiff  enough  also  to  prevent  improper  action  of  the 
teeth  due  to  springing  of  the  rim.  A  section  of  rim  between  two 
arms  may  be  considered  as  a  beam  fixed  at  the  ends  and  carrying 
a  load  at  the  middle,  the  value  of  which  is  W1  sin  6,  Fig.  151. 
Good  practice  makes  the  thickness  of  the  rim  at  least  1.25  /, 
where  t  is  the  thickness  of  the  tooth  on  the  pitch  line.  For  small 
gears  this  proportion  gives  ample  stiffness,  but  for  very  large 
gears  stiffening  ribs  are  also  sometimes  necessary.  In  many 
cases  the  thickness  should  be  sufficient  to  allow  of  dovetail- 


Shroud 


FIG.  159. 


FIG.  160. 


FIG.  161. 


ing  a  tooth  into  the  rim,  in  case  of  accidental  breakage  of  one 
or  more  teeth.  Gear  wheels  are  seldom  run  at  peripheral 
velocities  which  induce  dangerous  centrifugal  stresses.  The 
principles  governing  the  design  of  such  wheels  are  discussed, 
however,  in  Chap.  XV. 

The  arms  of  gear  wheels  may  be  treated  as  cantilevers,  assum- 

W 

ing  that  each  arm  carries  a  load  — ,  where  n  is  the  number  of 

n 

arms,  and  W  the  tangential  load.  Computations  for  strength 
of  either  arms  or  rims  must,  however,  be  considered  as  giving 
minimum  dimensions,  stiffness  being  the  prime  requirement,  and 
due  regard  must  be  paid  to  proportions  of  rim,  arms,  and  hub, 
to  minimize  shrinkage  stresses  due  to  cooling. 


392  MACHINE   DESIGN 

179.  Efficiency  of  Spur  Gearing.     The  experimental  data  on 
the  efficiency  of  spur  gearing  are  very  meagre.     Probably  the  best 
available  data  are  those  obtained  by  Mr.  Wilfred  Lewis,  for  details 
of  which  see  Trans.  A.  S.  M.  E.,  Vol.  VII.     His  investigation 
was  made  with  a  cut  spur  pinion  of  12  teeth  meshing  with  a  gear 
of  39  teeth.     The  circular  pitch  was  iX  inches  and  the  face  3^3 
inches.     The  load  varied  from  430  pounds  to  2,500  pounds  per 
tooth,  and  the  peripheral  speed  ranged  from  3  feet  to  200  feet 
per  minute.     The  measurements  included  the  friction   at   the 
teeth,  and  the  friction  at  the  bearings.     The  efficiency,  as  ob- 
served, varied  from  90  per  cent  at  a  velocity  of  3  feet  per  minute 
to  over  98  per  cent  at  200  feet  per  minute.     It  appears  that  the 
friction  at  the  teeth  is  a  small  part  of  the  loss  with  good  cut  gears, 
the  greater  portion  of  the  loss  being  at  the  journals.     The  effi- 
ciency of  bevel  gears  is  somewhat  less  than  that  of  spur  gears, 
on  account  of  the  axial  thrust,  which  induces  friction  between 
the  hub  of  the  gear  and  the  collar  at  the  supporting  bearing. 

HELICAL    OR  TWISTED   GEARING 

180.  General  Principles.     Suppose  a  spur  gear  to  be  cut  into 
n  small  sections  by  a  series  of  planes  perpendicular  to  the  axis  of 
rotation.     If  each  section  be  then  placed  a  proper  distance  ahead 
or  behind  the  adjacent  section, Fig.  161  (a),  it  is  evident  that  they 
may  be  so  arranged  that  some  one  section  is  just  coming  into 
contact  with  its  mating  section  when  the  nth  section  in  advance 
of  it  is  in  contact  at  the  pitch  point.     With  such  an  arrangement 
some  section  will  always  be  in  contact  near  the  pitch  point,  and 
there  will  always  be  approximately  n  points  of  contact  with  the 
mating  gear  between  the  pitch  point  and  the  point  which  marks 
the  beginning  of  tooth  action.     Since  the  action  of  gear  teeth  is 
smoothest  when  contact  is  near  the  pitch  point,  this  arrangement 
of  gearing  runs  more  quietly  and  smoothly  than  ordinary  spur 
gearing,  and  it  was  at  one  time  used  in  machine  tool  and  similar 
work  where  smooth  action  is  very  desirable. 

As  the  number  of  sections  is  increased,  the  total  width  of 
the  gear  remaining  the  same,  the  spacing  of  these  sections  being 


TOOTHED  GEARING  393 

kept  uniform  as  before,  the  form  of  the  stepped  tooth  approaches 
that  shown  in  Fig.  161  (c).  When  the  number  becomes  infinite 
the  teeth  become  helical  in  form,  and  contact  is  continuous  along 
that  portion  of  the  face  which  is  within  the  arc  of  contact.  It  is 
evident,  however,  that  since  the  relative  position  of  adjacent 
laminae  is  arbitrary,  and  may  follow  any  desired  law,  the  outline  of 
the  tooth  in  an  axial  direction  is  not  necessarily  helical,  but  may 
have  any  desired  shape;  although  these  teeth  are  most  usually 
made  helical,  this  form  being  more  practical  to  cut.  This 
form  of  gearing  is  also  known  as  twisted  gearing,  for  an  ob- 
vious reason.  The  action  of  such  gears  is  identical  with  that 
of  common  spur  gearing,  and  should  not  be  confused  with  that 
of  screw  gearing,  though  certain  limiting  forms  of  the  latter  are 
also  twisted  gears.  A  screw  gear  must  have  regular  or  uni- 
form helical  teeth,  while  a  twisted  gear  does  not  necessarily 
have  this  limitation. 

Since  the  pressure,  W,  between  mating  teeth  must  be  normal 
to  the  surface,  there  is  a  component,  Fig.  161  (c),  which  tends  to 
move  the  gear  in  an  axial  direction,  causing  end  thrust  on  the 
shaft  collars.  This  can  be  obviated  by  making  two  sets  of  helical 
teeth  on  each  gear,  one  right-hand  and  one  left-hand,  as  shown  in 
Fig.  162.  When  it  is  desired  to  use  cut  teeth  the  wheel  is  some- 
times made  in  two  parts  and  fastened  together,  or  the  wheel  may 
be  made  in  one  piece  and  the  two  sets  of  teeth  staggered  so  as  to 
allow  them  to  be  cut;  but  in  both  of  these  constructions  there  is 
some  loss  of  strength  due  to  the  absence  of  the  reinforcing  action 
of  teeth  cast  solid  as  in  Fig.  162.  Gears  of  this  type  are  also 
called  herring-bone  gears.  With  the  arrangement  shown  in  Fig. 
162,  care  must  be  used  that  the  alignment  in  an  axial  direction  is 
accurate,  or  end  play  must  be  provided  so  that  the  middle  plane 
of  both  gears  coincide;  otherwise  the  full  load  will  be  thrown 
on  one-half  the  gear  and  the  object  of  the  double  gear  defeated. 

181.  Strength  of  Twisted  Gears.  If  the  effective  load  which 
one  tooth  of  a  twisted  gear  transmits  to  its  mate  be  IF,  Fig.  161  (c), 
then  the  total  load  normal  to  the  face  is  W1  =  W  cosec  0.  If 
the  length  of  the  tooth  be  denoted  by  /,  and  the  breadth  of  the  gear 
by  £,  then  /  =  b  cosec  0.  Hence  the  load  per  inch  of  face  on 


394 


MACHINE  DESIGN 


Wl       W  cosec  0 

a  twisted  tooth  =—  =  — 

/          b  cosec  0 


W 

—  or  the  same  as    in    a 
b 


spur  gear  of  face  b.  This  would  be  strictly  true  if  all  points  in 
the  li  ae  of  contact  were  at  the  same  distance  from  the  axis  of  ro- 
tation as  in  a  spur  gear.  This  is  never  so,  in  twisted  gears,  the  line 
of  contact  always  extending  diagonally  across  the  tooth  face. 
The  error  due  to  this,  however,  is  small,  and  on  the  side  of  safety, 
and  it  may  be  assumed  that  the  load  per  inch  of  face  in 
twisted  gears  is  the  same  as  that  of  a  spur  gear  of  equal  width  and 
equally  loaded.  This  diagonal  distribution  of  the  load  across 
the  tooth  face,  decreases  the  lever  arm  of  the  force  which 
tends  to  break  the  tooth;  the  amount  of  decrease  depending 


FIG.  162. 


FIG.  163. 


FIG.  164. 


on  the  amount  of  twist  in  the  tooth.  If  the  twist  is  so  great 
that  when  the  end  in  advance  is  going  out  of  contact  the  other  end 
is  just  coming  into  contact,  the  line  of  contact  will  run  diagonally 
across  the  tooth  from  point  to  flank,  and  the  average  arm  of  the 
driving  force  will  be  about  one-half  the  height  of  the  tooth.  If 
the  twist  be  made  equal  to  the  pitch,  tooth  action  is  continuous 
at  every  point  of  the  arc  of  action  and  this  proportion  is  the  one 
most  used.  It  is  clear,  however,  that  the  assumption  often  made 
that  twisted  teeth  are  twice  as  strong  as  spur  teeth  of  the  same 
pitch  is  not  true  for  teeth  of  usual  proportions,  a  difference  of  25 
per  cent  being,  perhaps,  as  much  as  can  safely  be  assumed.  On 
account  of  continuous  tooth  action  and  consequent  smoother 
operation  in  twisted  gears,  the  effect  of  shock  is  lessened  some- 


TOOTHED   GEARING  395 

what.  Twisted  gears  have  been  used  with  success  on  heavy  wind- 
ing and  hoisting  engines,  the  teeth  being  often  rough  cast  and 
both  gear  and  pinion  half  shrouded,  making  a  very  strong  tooth. 

SCREW    GEARING 

182.  Forms  of  Screw  Gears.  When  the  axis  of  two  shafts  are 
not  parallel  and  do  not  intersect,  it  is  possible  to  lay  out  contact 
surfaces  on  which  gear  teeth  may  be  constructed  which  will  give 
line  contact.  Gears  of  this  kind  are  known  as  skew-bevel  gears. 
They  are  difficult  to  construct,  and  are  very  rarely  used.  If 
the  load  can  be  carried  on  point  contact,  pitch  cylinders  may  be 
described  on  the  axes,  Fig.  163,  and  on  these  surfaces  helical 
teeth  may  be  constructed  which  will  transmit  the  desired  motion. 
Such  gears  are  known  as  screw  or  spiral  *  gears,  the  latter  name 
being  really  a  misnomer.  While  the  teeth  of  such  gears  resemble 
those  of  helical  twisted  gears,  their  theory  and  action  are  quite 
different;  for,  in  addition  to  the  conjugate  rolling  and  sliding 
action,  as  in  spur  gears,  there  is  also  a  sliding  component  along 
the  elements  between  contact  surfaces.  The  action  of  screw 
gearing  is  very  smooth.  The  special  case  where  the  axes  are  at 
right  angles,  and  where  a  large  wheel  having  many  helical  teeth 
meshes  with  a  small  one  having  a  very  few  helical  teeth,  is  an  im- 
portant one  on  account  of  the  great  reduction  in  velocity  ratio 
that  may  thus  be  obtained.  This  last  arrangement  is  commonly 
known  as  a  worm  and  worm-wheel.  Fig.  165  illustrates  such 
a  worm  and  worm-wheel,  the  teeth  on  the  worm  wheel  being 
truly  helical  in  form  and  cut  at  an  angle  to  suit  the  worm  thread 
or  helix.  The  same  result  is  sometimes  obtained  by  using  a  plain 
spur  gear,  and  setting  the  axis  of  the  worm  at  the  proper  angle 
with  the  plane  of  the  gear.f  The  contact  in  these  cases  is  point 
contact,  and  on  the  worm  wheel  tooth  is  confined  to  points  in  a 
line  cut  from  the  working  surface  of  the  tooth  by  a  plane  passing 

*  For  a  full  discussion  of  the  methods  of  laying  out  and  producing  so-called 
spiral  gears,  see  a  "Practical  Treatise  on  Gearing,"  by  Brown  &  Sharpe  Mfg.  Co., 
and  also  "Worm  and  Spiral  Gearing,"  by  F.  A.  Halsey. 

t  A  highly  successful  form  of  this  arrangement  is  the  worm-and-rack  drives  on 
planing  machines,  first  used  by  Wm.  Sellers  &  Co. 


396  MACHINE   DESIGN 

through  the  axis  of  the  worm  at  right  angles  to  the  axis  of  the 
worm  wheel.  In  practice  the  point  of  contact  becomes  a  limited 
area.  The  advantage  of  this  form  of  worm  wheel,  like  all  spur 
gears,  is  that  the  teeth  can  be  cut  with  a  rotary  cutter,  and  patterns 
for  rough  cast  teeth  are  comparatively  easy  to  construct. 

It  is  possible,  however,  to  construct  a  worm  wheel  in  such  a 
manner  as  to  secure  line  contact,  as  in  spur  gearing.  Referring 
to  Fig.  164,  it  can  be  seen  that  when  the  single-threaded  worm 
shown  is  rotated  through  360°,  any  median  section  as  A  is  moved 
forward  an  amount  equal  to  the  pitch  of  the  worm  wheel  to  a 
position  B;  and  that  rotation  of  the  worm,  in  general,  is  equivalent 
to  a  translation  of  these  sections  backward  or  forward.  The 
action  is  equivalent  to  translating  a  rack  of  similar  proportions, 
and,  in  fact,  if  the  worm  itself  is  moved  axially  it  will  engage  with 
the  teeth  of  the  worm  wheel  in  the  same  manner  as  a  rack  does 
with  a  gear.  In  the  involute  system  of  gear  teeth  the  rack  has 
straight  sides,*  and  this  property  is  usually  taken  advantage  of 
in  making  worm  gearing,  since  a  worm  thread  of  such  a  cross- 
section  is  easily  machined.  The  sides  of  the  involute  rack  face 
are  at  right  angles  to  the  line  of  contact,  a  O  b,  Fig.  147,  and  hence 
the  inclination  of  the  sides  to  each  other  is  2  0,  Fig.  147,  and  in 
the  standard  system  2  0  =  29°.  If  other  planes  such  as  If  AT"  be 
passed  through  the  worm  and  worm  wheel  parallel  to  the  median 
plane  X  X,  Fig.  164,  it  will  cut  a  trapezoid  from  the  worm  some- 
what different  from  that  cut  by  the  median  plane.  The  rack- 
like  action  of  these  trapezoids  would,  however,  be  similar  to  those 
on  the  median  plane,  and  it  is  clear  that  the  shape  of  the  worm- 
wheel  tooth  in  the  plane  M  N  may  be  so  made  as  to  mesh  cor- 
rectly with  this  new  trapezoidal  section.  It  is  evident  that  if 
enough  such  sections  be  taken,  a  complete  tooth  outline  may  be 
formed  that  will  give  line  contact  with  a  worm  across  its  full  face. 
It  is  evident  also  that  any  other  form  cf  worm  thread  may  be 
similarly  treated. 

The  preceding  discussion  demonstrates  the  possibility  of 
line  contact  in  screw  gearing,  and  suggests  a  method  by  which 

*  See  "  Kinematics  of  Machinery,"  by  John  H.  Barr,  page  125. 


TOOTHED   GEARING 


397 


the  teeth  of  such  gearing  could  be  drawn,  and  hence  constructed. 
There  is  no  practical  value  in  actually  making  such  drawings;  but 
teeth  having  this  property  of  line  contact  are  automatically  pro- 
duced by  what  is  known  as  the  bobbing  process.  A  worm  wheel 
of  tool  steel  is  made  of  the  exact  form  of  the  desired  worm.  This 
worm  is  made  into  a  cutter  by  cutting  flutes  across  the  face  as  in 
Fig.  1 68.  This  is  known  as  a  hob;  and  when  hardened  and 
tempered  it  is  used  as  a  milling  cutter.  The  wheel  blank,  which 
has  been  turned  to  correspond  to  the  outside  of  the  teeth,  is 
mounted  in  a  gear  cutter,  or  a  special  hobbing  machine,  and  the 


FIG.  165. 


FIG.  166. 


hob  is  also  mounted  in  correct  relation  to  the  wheel,  but  with  the 
axes  of  the  wheels  a  little  greater  distance  apart  than  the  required 
final  distance.  The  hob  is  then  rotated  and  at  the  same  time  fed 
toward  the  worm  wheel  till  the  proper  distance  between  the  axes 
is  reached,  thus  cutting  the  teeth  in  the  worm  wheel  in  a  very 
accurate  manner.  Sometimes  the  wheel  is  caused  to  rotate  simply 
by  the  action  of  the  hob,  but  much  better  results  are  obtained  if 
it  is  driven  positively,  with  the  proper  velocity  ratio,  from  the 
cutter  spindle  by  means  of  positive  gearing.  In  heavy  work  the 
teeth  of  the  wheel  are  roughed  out  or  " gashed"  before  hobbing. 


398  MACHINE  DESIGN 

Fig.  1 66  shows  a  worm  wheel  which  has  been  hobbed,  and  its 
mating  worm.  Fig.  167*  shows  a  form  of  wheel  occasionally 
used  where  the  wheel  is  sometimes  rotated  by  hand  or  when  the 
projecting  teeth  are  undesirable.  Such  wheels  may  be  hobbed, 
but  are  usually  cut  by  the  approximate  method  shown  in  Fig.  169, 
where  a  cutter  is  fed  radially  inward  toward  the  axis  of  the  worm 
wheel,  producing  what  is  known  as  a  drop-cut  wheel.  In  the 
Hindley  worm  the  pitch  line  of  the  worm  is  curved  to  coincide 
with  the  pitch  line  of  the  wheel,  thus  obtaining  contact  on  several 
teeth  at  the  same  time.f 

183.  Velocity  Ratio  of  Worm  Gearing.  The  axial  advance 
per  turn  of  the  worm  thread  is  called  the  lead.  Thus  in  Fig.  164 
the  lead  of  the  single-threaded  worm  shown  is  the  distance,  parallel 
to  the  axis,  from  any  point  on  the  tooth  section  A,  to  a  correspond- 
ing point  on  the  section  B,  and  is  equal  to  the  circumferential 
pitch  of  the  worm  wheel.  If  the  worm  were  double-threaded  the 
lead  would  be  twice  this  amount,  or  equal  to  the  distance  between 
corresponding  points  on  A  and  C,  and  would  then  be  twice  the  pitch 
of  the  worm  wheel.  The  lead  of  the  triple-threaded  worm  would 
be  three  times  the  pitch,  and  so  on.  If  a  single-threaded  worm 
makes  one  revolution,  a  tooth  of  the  worm  wheel  is  moved  a 
distance  equal  to  the  pitch.  In  the  case  of  a  double-threaded 
worm  the  tooth  would  be  moved  twice  the  pitch;  and  in  general 
if  N  be  the  number  of  teeth  in' the  worm  wheel,  and  n  the  number 

,   .  angular  velocity  of  worm         N 

of  threads  on  the  worm,  then,  -  — - — •=-*—  — ;  =  — . 

angular  vel.  of  worm-wheel       n 

Evidently  a  very  great  velocity  ratio  is  possible  with  a  compara- 
tively small  worm-wheel.  It  is  to  be  especially  noted  that  the 
angular  velocity  ratio  is  independent  of  the  diameter  of  the  worm. 
The  pitch  of  the  worm  wheel,  which  must  be  decided  upon  by 
consideration  of  the  strength  of  the  teeth,  fixes  the  radius  of  the 
worm  wheel  for  a  given  number  of  teeth;  but  the  radius  of  the 
worm  may  then  be  varied  to  suit  other  conditions. 

*  Figs.  165,  166,  and  167  are  reproduced  from  Browne  &  Sharpe's  "Treatise 
on  Gearing." 

f  See  "Worm  and  Spiral  Gearing,"  by  F.  A.  Halsey. 


TOOTHED   GEARING 


399 


184.  Efficiency  of  Worm  Gearing.  The  general  expressions 
for  the  efficiency  of  screws,  deduced  in  Art.  54  of  Chap.  VII, 
apply  also  to  worm-gearing.  Since  the  worm  thread  is,  usually, 
a  so-called  angular  thread,  equation  13  (a)  of  that  Article  would 
strictly  apply.  However,  the  inclination  of  the  face  of  worm 
threads  is  so  small  that  the  error  introduced  in  using  the  simpler 
equations  (9)  and  (10)  of  that  article,  which  were  deduced  from 
the  square  thread,  is  small.  These  equations  show  that  the 
efficiency  of  all  screw  gears  is  a  function  of  the  angle  which  the 
thread  makes  with  a  plane  perpendicular  to  the  axis,  and  of 


FIG.  167. 


FIG.  168. 


FIG.  169. 


the  coefficient  of  friction,  assuming  that  the  coefficient  of  fric- 
tion at  the  thrust  collar  is  the  same  as  at  the  tooth. 

One  of  the  most  valuable  contributions  to  this  subject  is  the 
experimental  work  of  Mr.  Wilfred  Lewis.*  The  full  lines  in 
Fig.  170  have  been  plotted  from  the  diagram  on  which  he  has 
summarized  his  results.  They  show  clearly  the  increase  of 
efficiency  with  increase  of  thread  angle  at  all  velocities.  They 
also  show  a  remarkable  agreement  with  the  theoretical  equations 
of  Art.  54.  The  dotted  curve  is  reproduced  from  curve  (2)  of 
Fig.  52,  and  its  close  agreement  with  Mr.  Lewis'  curves  is  to  be 
noted.  This  dotted  curve  was  plotted  for  a  value  of  /*  =  0.05. 


*  Trans.  A.  S.  M.  E.,  Vol.  VIl,  page  297. 


400 


MACHINE   DESIGN 


Mr.  Lewis'  calculated  average  value  of  this  coefficient  for  a 
velocity  of  20  feet  per  minute  is  0.059  and  for  10  feet  per  minute 
0.074.  Curves  (4)  and  (5)  in  Fig.  52  may,  therefore,  be  taken 
as  supplementary  to  those  in  Fig.  170,  and  may  be  used,  as  they 
were  intended,  for  designing  slow-moving  and  poorly  lubricated 
screws.  A  theoretical  curve  plotted  from  equation  (9),  Art.  54, 
with  a  value  of  /JL  =  0.014  (which  would  be  obtained  only  at 
high  speeds),  will  coincide  very  closely  with  curve  i,  Fig.  170. 


aoo 


90 


80 


30 


10  15  20  25 

Lead  Angle  in  Degrees 

FIG.  170. 


35 


15 


This  coincidence  is  closer  than  might  be  expected  from  the 
nature  of  the  problem  and  the  assumptions  on  which  equation 
(9)  is  based.  Mr.  Lewis'  value  of  p  for  these  velocities  *  (200 
feet  per  minute)  ranged  from  0.026  to  0.015,  his  average  value 
being  0.02. 

Mr.  Halsey  f  has  examined  the  design  of  a  number  of  success 

*  Velocity  here  means  velocity  of  rubbing  at  the  point  of  contact  between  worm 
and  worm  wheel. 

t  See  "Worm  and  Spiral  Gearing,"  page  38. 


TOOTHED    GEARING  401 

ful  and  unsuccessful  worms  used  for  transmitting  power  and 
found  that  every  worm  among  those  examined  whose  lead  angle 
was  greater  than  12°-  30'  was  successful,  and  every  worm  whose 
lead  angle  was  less  than  9°  was  unsuccessful,  and  quotes  Mr.  James 
Christie,  who  has  had  considerable  experience  with  this  form  of 
gearing,  as  giving  17°  —  15'  as  the  lower  limit  for  successful  design, 
which  still  further  corroborates  the  general  theory  given.  It  is 
to  be  noted,  on  the  other  hand,  that  there  is  little  to  be  gained  in 
using  a  pitch  angle  above  30°,  the  increase  in  efficiency  being 
very  small,  while  the  side  thrust  on  the  wheel  is  increased.  It 
is  not  to  be  understood  that  it  is  never  proper  to  design  a  worm 
with  a  lead  angle  less  than  9°;  for  there  are  many  cases,  not 
primarily  for  power  transmission,  and  where  the  velocity  is  low, 
in  which  worms  of  less  pitch  are  not  only  effective  but  neces- 
sary. In  Mr.  Lewis'  experiments  the  worms  ran  in  a  bath  of 
oil,  and  the  efficiencies  given  include  journal  friction,  the  thrust 
being  taken  at  the  end  of  the  worm  shaft  by  a  loose  brass 
washer  running  between  two  hardened  and  ground  steel  washers 
(see  Art.  104). 

The  effect  of  the  velocity  of  rubbing  on  the  coefficient  of 
friction  of  imperfectly  lubricated  surfaces,  was  noted  in  Art.  32, 
and  Fig.  17  of  that  article  indicates,  in  a  general  way,  what  may 
be  expected  with  sliding  surfaces:  all  experimental  results  going  to 
show  that  the  lowest  coefficient  was  obtained  at  about  200  feet  per 
minute.  Mr.  Lewis,  as  the  result  of  his  work,  fixes  200  feet  per 
minute  as  the  point  of  maximum  efficiency  of  worm  gearing, 
which  is  in  perfect  accord  with  the  general  theory  of  lubrication. 
The  surfaces  of  worm  gearing,  although  running  in  an  oil  bath, 
must,  from  the  nature  of  the  contact,  be  classified  as  imperfectly 
lubricated  surfaces.  An  increase  of  velocity  may,  up  to  a  certain 
limit,  decrease  the  coefficient  of  friction,  but  it  is  not  possible  at 
any  speed,  with  the  small  amount  of  surface  contact  obtainable 
in  screw  gearing,  to  create  a  true  oil  film  so  that  the  load  would 
be  fluid-borne  (Art.  33). 

185.  Limiting  Pressures  and  Velocities  in  Worm  Gearing.  It 
was  stated  in  the  last  two  articles  that  the  best  results  are  obtained 
from  worm  gearing  when  the  rubbing  velocity  is  about  200  feet 
26 


402 


MACHINE   DESIGN 


per  minute  and  the  lead  angle  not  less  than  12° -30'.  It  is  not 
always  possible,  however,  to  keep  the  design  within  these  limits. 
Thus  in  order  to  obtain  mechanical  advantage  (see  Art.  64),  it  may 
be  necessary  to  use  a  worm  with  a  very  small  lead  angle,  and 
kinematic  requirements  may  necessitate  a  much  higher  velocity 
than  200  feet  at  the  pitch  line. 

The  allowable  axial  load  that  may  be  applied  to  a  worm  under 
varying  velocities  has  not  been  very  accurately  determined,  the 
law  undoubtedly  being  complex  (see  Art.  32).  Enough  experi- 
mental work  has  been  done,  however,  to  show  that  the  pressure 
varies,  approximately,  inversely  with  the  velocity;  or  the  law 
may  be  roughly  expressed  as  W  V  =  K,  where  W  =  the  axial  load 
on  the  worm,  V  =  the  velocity  of  rubbing  in  feet  per  minute,  and 
K  =  a  constant  to  be  determined  by  experiment  (see  also  Art. 
98).  In  Lewis'  experiments,  made  on  cast-iron  worms  and  worm 
wheels,  running  in  an  oil  bath,  it  was  found  that  the  limiting  value 
of  K,  i.e.,  where  cutting  began,  was  about  1,500,000.  Smith  and 
Marx*  quote  corresponding  pressures  and  velocities,  attributed 
to  Stribeck,  obtained  with  hardened  steel  worm  and  bronze 
worm  wheel  running  in  an  oil  bath,  which  give  an  average  allow- 
able value  of  690,000  for  K.  Bach  and  Roser;  experimenting  with 
soft-steel  worms  and  bronze  worm  wheels,  succeeded  in  carrying 
a  pressure  of  800  pounds  at  a  velocity  of  1,700  feet  per  minute, 
which  gives  K  =  1,360,000.  It  would  appear,  therefore,  that  for 
average  conditions  and  bath  lubrication  of  the  worm  it  will  be 
safe,  for  velocities  up  to  1,500  feet  per  minute,  to  take 

W  V  =  750,000 (14) 

The  above  discussion  has  reference  to  worms  as  ordinarily  con- 
structed with  straight-sided  threads.  Mr.  Robert  Bruce f  has 
shown  that  if  the  sides  of  the  worm  are  made  concave  a  much 
greater  load  may  be  carried.  With  improved  threads  of  this  form 
he  has  succeeded  in  carrying  25  tons  at  a  velocity  of  120  feet  per 
minute,  corresponding  to  K  =  6,720,000.  This  great  gain  is  due, 

*  "  Machine  Design,"  page  301. 

t  Proceedings  of  Institution  of  Mechanical  Engineers  (British),  page  57  of 
the  year  1906. 


TOOTHED   GEARING  403 

without  doubt,  to  the  improved  lubrication  obtained  by  what 
practically  amounts  to  surface  contact,  between  the  mating  convex 
and  concave  surfaces  of  the  teeth. 

1 86.  Design  of  Worm  Gearing.  In  general,  the  strength  of 
the  worm  exceeds  the  strength  of  the  teeth  in  the  worm-wheel; 
and  where  the  worm  is  made  of  a  harder  material,  which  is  the 
usual  case,  the  wear  is  greatest  on  the  worm-wheel  teeth.  It 
is  usually  sufficient,  therefore,  to  design  the  wheel  teeth  alone, 
considering  them  as  simple  spur  gear  teeth  as  in  Art.  173.  In 
the  case  of  rough-cast,  or  drop-cut  teeth,  it  must  be  assumed 
that  the  entire  load  is  carried  by  a  single  tooth;  but  in  hobbed 
gearing  it  is  safe  to  assume  that  the  load  is  distributed  between 
two,  or  even  three,  teeth,  depending  on  the  number  of  teeth  in  the 
wheel. 

Example.  Design  a  worm  gear  to  connect  two  shafts  which 
are  n  inches  apart,  and  to  transmit  yX  H.-P.  The  velocity  ratio 
is  to  be  20  to  i,  the  worm  shaft  is  to  make  320  R.P.M.,  the  lead 
angle  is  not  to  be  less  than  15°,  and  the  worm  wheel  is  to  be  cut 
with  a  hob. 

The  solution  of  problems  in  worm  gearing  must,  generally, 
be  tentative.  If  the  velocity  ratio  is  to  be  20  to  i,  the  worm-wheel 
will  have  20,  40,  or  60  teeth,  depending  on  whether  the  worm  is 
single-,  double-,  or  triple-threaded.  It  is  difficult  to  obtain  a  high 
lead  angle  with  a  single-threaded  worm  without  making  a  very 
large  thread,  therefore  a  trial  assumption  will  be  made  with  a 
triple-threaded  worm,  and  60  teeth  in  the  wheel.  Twenty  inches 
may  be  taken  as  a  trial  diameter  for  the  wheel,  and  the  trial  pitch 
circumference  will  therefore  be  63  inches  approximately.  If  the 
circumferential  pitch  be  taken  as  one  inch,  the  lead  of  the  worm 
thread  will  be  three  inches,  and  can  therefore  be  easily  cut  in  a 
lathe.  The  corrected  circumference  of  the  wheel  will  then  be 
60",  corresponding  to  a  pitch  diameter  of  19.11".  The  pitch 
diameter  of  the  worm,  with  the  given  distance  between  centres, 

will  be  2.9";   hence  the  tangent  of  the  lead  angle  = 


71  X  2.9 
0.33,  or  the  lead  angle  is  18°  —  15',  which  is  an  efficient  angle. 


404  MACHINE   DESIGN 

The  number  of  revolutions  per  minute  of  the  worm  wheel  will 

be  - —  =  1 6.     Hence  the  velocity  of  the  worm  wheel  at  the  pitch 
20 

line  =  =  80  feet  per  minute.     The  total    axial   thrust 

12 

on  the  worm  will  be  — — 0          -  =  3,100  pounds.     The  velo- 

oO 

city  of  rubbing  equals  the  length  of  one  turn  of  the  worm 
thread  multiplied  by  the  number  of  revolutions  per  minute,  or 

TT  x  2.9  X  320  p  X  2.9  X  320 

V=  -  -  = ~ —  =  2SS  ft. per  minute. 

(cosiS'  —  15")  X  12  0.95  X  12 

The  product  of  velocity  and  axial  pressure  on  the  worm  =  255  X 
3,100  =  790,000  which  by  equation  (14)  is  a  safe  value,  although 
somewhat  high. 

The  load  may  be  considered  as  distributed  between  two  teeth, 
and  each  tooth  will  have  a  face  or  length  at  the  root  at  least  equal 
to  the  pitch  of  the  worm  (see  Fig.  164),  or  say  2.75".  Hence 

the  load  per  inch  of  face  of  tooth  =  -  =  560  pounds. 

2X2.75 

From  the  diagram,  Fig.  158,  it  is  seen  that  this  load  corresponds 
to  a  fibre  stress  of  about  5,000  pounds  per  square  inch  with  i 
inch  circular  pitch.  From  equation  (10)  of  Art.  175,  however, 
it  is  seen  that  for  the  velocity,  80  feet,  the  allowable  stress  is  7,000 
pounds,  hence  the  tooth  has  an  excess  of  strength  to  provide 
against  the  wear,  which  falls  heaviest  on  the  worm  wheel. 

From  curve  (i),  Fig.  170,  it  is  found  that  the  efficiency  is  about 
90  per  cent;  hence  the  horse-power  which  must  be  supplied  to 

furnish  7.5  H.-P.at  the  worm-wheel  shaft  will  be  —  —  =  8.4H.-P. 

0.90 

=  277,200  foot-pounds  per  minute,  or  866  foot-pounds  per  revolu- 
tion of  the  worm.  The  torque  J1,  which  must  be  applied  to  the 

worm-wheel  shaft,  will  be  T  =  -  -  =  1,650  inch-pounds. 

The  depth  below  the  pitch  line  of  a  standard  tooth  of  one  inch 
circular  pitch  is,  from  Table  XXV,  0.3857  inches;  therefore  the 


TOOTHED   GEARING  405 

diameter  of  the  worm  at  the  root  of  the  thread  =  2.9  —  (2  X  0.3857) 
=  2.13",  and  from  equation  E,  page  94,  the  torsional  stress  ps  = 

i6T        16  X  1,650 

— js  = ~f ^  =  °5°  pounds  per  square  inch,  which  is  very 

Trd3        TT  x  (2.I3)3 

low.  The  design  may,  therefore,  be  considered  satisfactory  if 
the  worm  is  to  be  cut  integral  with  the  shaft.  If,  however,  it  is 
to  be  bored  out  and  fitted  over  the  shaft,  further  calculation  as  to 
the  strength  of  the  shaft  which  may  be  fitted  is  necessary. 

187.  Thrust  Bearings  for  Worms.  An  important  frictional 
loss  in  worm  gearing  occurs  in  the  thrust  bearing,  which  therefore 
deserves  special  attention.  The  general  discussion  in  Art.  104 
applies  in  this  case.  The  type  of  bearing  shown  in  Fig.  88  is 
much  used,  and  of  late  ball  bearings  have  met  with  considerable 
success  in  such  places. 


CHAPTER     XV 
FLYWHEELS  AND  PULLEYS 

1 88.  Capacity  of  Flywheels. — There  are  two  distinct  types  of 
flywheels;  namely,  those  whose  sole  function  is  to  absorb  and 
redistribute  energy,  as  noted^in  Articles  2,  4,  and  6.2,  and  those 
which  also  act  as  a  pulley  or  band  wheel  and  transmit  power 
continuously.  When  a  flywheel  is  attached  to  a  train  of  mechan- 
ism in  which  the  supply  of  energy  varies  it  tends  to  absorb  any 
excess  energy,  thus  having  its  velocity  increased.  When  the 
work  to  be  done  is  in  excess  of  the  energy  supply,  the  wheel  tends 
to  furnish  the  deficiency  at  the  expense  of  its  kinetic  energy, 
with  a  resulting  reduction  of  velocity.  Flywheels,  therefore,  to  be 
effective  must  vary  in  velocity;  the  allowable  amount  of  variation 
depending  on  the  conditions  of  the  case.  Thus  in  engines 
driving  electric  generators,  the  variation  from  normal  speed 
may  be  limited  to  one-half  of  one  per  cent,  or  less,  while  in  such 
machines  as  punching  machines,  the  variation  may  be  as  great 
as  twenty  per  cent. 

If  W  be  the  weight  in  pounds  of  a  body  moving  with  a  velocity 
of  v  feet  per  second,  then  the  kinetic  energy  in  foot-pounds  which 

W  v2 
the  body  possesses  is  K  =  -    —  where  g  =  32.2.     If  the  velocity 

of  the  body  be  changed  from  v1  to  v2,  the  change  in  kinetic  energy 
is  the  work  which  the  body  will  do,  or  the  energy  it  will  absorb, 
depending  on  whether  its  velocity  is  decreased  or  increased.  If, 
then,  the  work  to  be  done  or  the  energy  to  be  absorbed  with  a 
given  change  in  velocity  is  known,  the  necessary  weight  of  the 
body  may  be  found;  for  if  K1  be  the  kinetic  energy  of  the  body 
when  moving  with  a  velocity  vlt  and  K2  be  the  kinetic  energy  at  a 
velocity  v2J  then  the  energy  delivered  or  absorbed  during  a  change 
of  velocity  is 

406 


FLYWHEELS   AND   PULLEYS  407 

W  V  2         W 


If  the  body  is  rotating  around  a  fixed  axis,  the  velocities  of  differ- 
ent points  in  the  body  vary  as  the  distance  of  these  points  from  the 

W 

axis.     For  this  case  the  kinetic  energy  of  the  body  is  -  -  p2  ^ 

where  p  is  the  radius  of  gyration,  and  <a  the  angular  velocity. 
Hence  for  rotating  bodies  equation  (i)  may  be  written 

E  =  K,  -  K2  =  ^  K2  -  -22)       ...      (2) 

W  p2 
or  since  --  =  /,  the  moment  of  inertia  *  of  the  body, 

o 

E  =  K,-K2  =  -2  K2  -  -22)  ....      (3) 

In  all  cases  of  flywheel  design  the  effect  of  the  hub  may  be 
neglected,  and  in  nearly  all  cases  the  effect  of  the  arms  is  so  small 
as  to  be  negligible,  and  the  rim  only  need  be  considered.  When 
such  is  the  case  it  is  sufficiently  accurate  to  take  the  mean  radius 
of  the  rim  R  as  the  radius  of  gyration,  and  equation  (2)  becomes 
identical  with  equation  (i)  since,  in  general,  R  u>  =  v.  In  the  case 
of  wheels  with  many  heavy  arms,  or  heavy  disc  wheels,  and  where 
it  is  desirable  to  compute  the  inertia  effect  of  the  wheel  closely, 
as  in  direct  driving  of  electric  generators,  equation  (2)  or  equation 
(3)  is  applicable.  In  the  case  of  a  wheel  with  arms  whose  sides 
are  parallel,  or  nearly  so,  it  is  to  be  noted  that  the  square  of  the 
radius  of  gyration  of  the  arms  or  o2  is  very  nearly  equal  to  %  R2. 
Hence  for  this  case,  if  W  be  the  weight  of  the  rim  and  Wa  'the 
total  weight  of  the  arms 

E  =  K,  -  K2  =        (W  +  X  W)  «  -  -22)  -(4) 


Example  (i).     A  punching   machine  is  to  make  30  strokes 
per  minute  and  is  to  punch  holes  y  in  diameter  in  steel  plate  y2" 

*  The  student  should  distinguish  clearly  between  the  moment  of  inertia  of  a 
solid  body  and  the  moment  of  inertia  of  an  area.    See  Church's  "Mechanics,"  Art 

W  R2 
86,  page  91.     In  the  case  of  a  circular  disc  I   =  -  . 


408 


MACHINE  DESIGN 


thick.  Since  the  machine  may  be  used  for  shearing  also,  it  should 
be  capable  of  punching  a  hole,  or  of  doing  the  equivalent  amount 
of  work  in  shearing,  at  every  stroke  of  the  punch,  continuously. 
The  belt  speed  is  to  be  about  600  feet  per  minute,  and,  from 
existing  machines  of  the  same  type,  it  is  known  that  the  efficiency 
will  not  exceed  85  per  cent.  It  is  required  to  find  the  cross- 
section  of  the  flywheel  rim. 

Let  Fig.  171  represent  the  machine  under  discussion.  The 
mechanism  in  the  head,  A,  is  a  slotted  crosshead;*  so  that  the 
punch  M  moves  with  harmonic  motion.  Let  the  diagram,  Fig. 
171  (a),  represent  the  path  of  the  pin  F,  in  the  crosshead.  When 


FIG.  171  (a). 


FIG.  171. 


the  pin  is  at  b  the  punch  enters  the  plate,  and  emerges  from  the 
lower  face  of  the  plate  when  the  pin  is  at  c.  When  the  pin  is  at  d 
the  punch  is  at  its  lowest  position,  and  has  entered  the  die  X";  at  e 
the  punch  is  withdrawn  from  the  plate  and  at  /  is  at  its  highest 
position.  The  pin,  therefore,  moves  through  an  angle  of  30° 
while  the  work  of  punching  is  being  performed. 

"The  preliminary  layout  also  shows  that  the  diameter  of  the 
driving  pulley  N  should  not  exceed  18",  and  the  mean  diameter 
of  the  flywheel  rim  should  not  exceed  42".  A  preliminary 
estimate  also  fixes  the  ratio  of  the  diameter  of  the  pinion  B  to 
the  diameter  of  the  gear  C  as  i  to  6;  hence  the  driving  shaft  will 
make  30  X  6  =  180  R.P.M.  The  circumference  of  the  driving 

*  See  "Kinematics  of  Machinery,"  by  John  H.  Barr,  page  184. 


FLYWHEELS  AND   PULLEYS  409 

belt  speed       600 
pulley  =  -  — —  -  =  —JT-  =  3.34  feet  and,  therefore,  the  diam- 

IoO  loO 

eter  of  the  driving  pulley  will  be  13"  which  is  well  within  the 
limit  set.  The  machine  makes  an  energy  cycle  (see  Art.  4) 
every  stroke  of  the  punch,  or  every  six  revolutions  of  the  driving 
shaft.  While  the  hole  is  being  punched  the  flywheel  is  giving  up 
energy  to  assist  the  belt,  and  during  the  remainder  of  the  cycle 
the  belt  withdraws  the  punch  from  the  sheet,  and  restores  the 
wheel  to  normal  speed. 

The  greatest  pressure  which  the  punch  must  exert  on  the  plate 
will  be  at  the  beginning  of  the  punching  operation,  and  will  be 
equal  to  the  area  of  the  metal  in  shear  multiplied  by  the  shearing 

resistance;    orP  =  7rX-X-X  60,000*  =  70,800  Ibs.     If  the 

belt  had  to  exert  this  effort  unaided,  it  would  have  to  be  of  double 
leather  15  inches  wide;  hence  the  need  of  a  flywheel.  As  the 
punch  passes  through  the  plate  the  shearing  resistance  decreases, 
until  it  becomes  zero  as  the  punch  passes  out.  The  average 
pressure  may  therefore  be  taken  as  half  the  maximum,  and  the 

70,800        i        i 
total  work  performed  in  punching  is-         -X  —  X  —  =  1480 

ft.-lbs.  The  work  of  withdrawing  the  punch  from  the  sheet  is 
small  and  may  be  considered  as  part  of  the  frictional  loss.  Since 
the  efficiency  of  the  machine  is  85  per  cent  the  belt  must  supply 

— =  1740  ft.-lbs.  every  energy  cycle.     The  energy  delivered 

0.85 

by  the  belt  per  cycle  is  the  product  of  the  difference  of  the  belt 
tensions  (Tl  —  T2)  multiplied  by  the  distance  through  which  the 
belt  moves  (see  Art.  131).  Since  the  speed  of  the  belt  is  600  feet 
per  minute,  and  the  time  of  the  cycle  is  -fa  of  a  minute,  the  belt 
moves  a  distance  =  -^°/  =  20  feet  per  cycle.  Hence  ( 3T,  —  T2)  20 
=  1740  or  7\  -  T2  =  87  Ibs.  The  effect  of  centrifugal 
force  may  be  neglected  at  this  belt  speed,  hence  equation  (8), 
Art.  131,  gives/  =  o.6/  for  an  arc  of  contact  of  180°  and  /*  = 

*  The  shearing  strength  of  a  plate  in  punching  is  about  equal  to  its  tensile 
strength. 


MACHINE  DESIGN 

0.3,  where/  =  the  effective  pull  per  inch  of  width  of  belt  and  /  = 
tension  per  inch  of  width  of  belt  on  the  tight  side.  In  this  class 
of  machinery  where  excessive  slipping  of  the  belt  is  sure  to  occur, 
/  should  be  taken  at  not  more  than  40  Ibs.  per  inch  of  width; 
whence  /  =  0.6  X  40  =  24  Ibs.,  and  the  width  of  the  belt  = 

S7 

— L  =  <i%  inches. 

24 

Since  the  pin,  Fig.  171  (a),  moves  through  30°,  or  TV  of  a 
revolution,  during  the  operation  of  punching,  and  since  the 
punch  makes  30  strokes  per  minute,  the  .time  consumed  during 

the  operation  of  punching  will  be    —  X  —  =  — —  of  a  minute. 

12       30       360 

The   belt,    therefore,    moves    600  X  -g-io-  =1.7    ft.    during    the 
operation  and  supplies  87  X  1.7  =  150  ft.-lbs.,  leaving  1750  - 
150  =  1600  ft.-lbs.  to  be  supplied  by  the  flywheel.     The  driving 
shaft  makes   180  R.P.M.,  or  3    revolutions  per    second    and, 
therefore,   since   the   mean   radius   of   the   wheel  =21    inches, 

2    X   7T   X    21    X3 

v±  =  -  -  =  33  feet  per  second,. 

The  allowable  variation  in  velocity  may  be  taken  as  10  per  cent, 
hence  v2  =  33  X  0.90  =  30  feet  per  second.  Therefore,  neglect- 
ing the  effect  of  the  hub  and  arms,  the  weight  of  the  rim  is  from 
equation  (i) 

_      2gE      _  2  X  32.2  X  i ,600 

vi2  ~"  v*  332  ~~  3°2 

One  cubic  inch  of  cast  iron  weighs  0.26  Ibs.,  hence  the  number  of 

cubic  inches  in  the  rim  will  be         .  =  2,100.     The  mean  cir- 

0.26 

cumference  of  the  rim  =  71X42  =  132  inches;    therefore  the 

2,100 

cross-section  =  =  16  square  inches,  or  a  section  3-H  inches 

132 

wide  by  5  inches  deep  may  be  selected. 

In  the  above  example  it  is  quite  easy  to  compute  the  amount 
of  energy  to  be  redistributed,  from  the  conditions  of  the  problem. 
In  the  more  general  case  this  cannot  be  done  so  readily,  and 
methods  such  as  those  outlined  in  Articles  5  and  6  of  Chapter 


FLYWHEELS   AND   PULLEYS  411 

II  must  be  employed,  where  the  diagram  representing  work  to  be 
done  is  superimposed  upon  that  representing  the  energy  supplied, 
in  such  a  manner  that  the  excess  and  deficiency  may  be  measured. 
Care  should  also  be  exercised  that  the  solution  covers  a  complete 
energy  cycle  in  order  that  the  solution  may  be  based  on  the  greatest 
excess  or  deficiency.  Thus  in  Fig.  2  (d)  the  flywheel  redistributes 
energy  on  both  strokes,  but  the  maximum  excess  of  effort  is 
represented  by  K  and  not  by  Kt.  Again  in  such  machines  as 
internal-combustion  engines  with  "hit-and-miss"  governors, 
giving  a  very  variable  energy  supply,  the  design  of  the  flywheel 
may  have  to  be  based  on  a  hypothetical  performance  of  the 
engine  covering  a  number  of  successive  strokes,  or  in  other  cases 
may  be  based  on  empirical  constants  which  are  the  result  of  ex- 
perience. 

In  some  machinery,  such  as  steam  engines,  it  is  desirable 
to  limit  the  variation  in  velocity  to  a  definite  amount  both  above 
and  below  the  mean  velocity.  If  v  be  the  mean  velocity,  v1  and 
v2  the  maximum  and  minimum  velocities  respectively,  then  it  is 
sufficiently  accurate  for  most  work  to  take  v  =  K  fyi  +  v2), 
but  the  true  relation  between  these  quantities  depends  on  the 
manner  in  which  the  vejocity  changes.  For  very  exact  work,  as 
in  parallel  operation  of  alternating  generators,  it  may  be  necessary 
to  take  this  into  account.* 

Example  (2).  Find  the  weight  of  a  cast-iron  flywheel  neces- 
sary to  limit  the  speed  of  the  engine  discussed  in  Art.  5  to  a  total 
variation  of  not  more  than  o.oi,  i.e.,  0.005  above  or  below  the 
mean  speed.  The  wheel  is  also  to  act  as  a  belt  wheel,  and  the 
belt  speed  is  to  be  about  4,000  feet  per  minute.  The  sides  of 
the  arms  are  to  be  parallel  and  their  effect  is  to  be  considered. 

Since  the  engine  makes  160  R.P.M.,   the  circumference  of 

the  wheel   =  -      -  =  25  feet.      Therefore  the  diameter  of  the 
1 60 

wheel  =  8  feet,  which  may  be  taken  without  great  error  as  a 
mean  diameter  of  the  wheel  rim,  since  the  thickness  of  the  rim 
will  not  be  great.  The  face  of  the  wheel  should  be  at  least  16 

*  See  a  paper  by  I.  J.  Astrom,  Trans.  A.  S.  M.  E.,  Vol.  XXII,  page  972. 


4I2  MACHINE   DESIGN 

inches  wide  to  give  the  necessary  width  of  belt.     A  preliminary 
layout  gives  6  arms,  each  weighing  170  pounds,  or  W&  =  170  X  6  = 

1,020.      Let  n  =  the  revolutions  per  second  =  —  and  R  =  the 

radius  =  4  feet;   from  Art.  5,  E  =  4,620  foot-pounds.     Then  «> 

160 

=  27t  n  =  2n  X  T~  =  l6-8  radians. 
60 

CD      -J-    (It 

Now   co  =  -       — -  and   hence   from   the   conditions   of   the 

2 

problem,    ^  =  i.oo5«>   =   1^005   X   16.8  =   16.88 ;    and  <«2    = 
0.995  "*  =  I^-7-     Therefore 

(w*  -  w22)  =  (i6.882  -  i6.72)  =6.1 

From  Equation  (4),  (W  +  1A  ^J  =  ^T, 


2  gE  WA       2X32.2X4,620      1, 

"^-j-w-vry     16x6.1 

Therefore  PF  +  Wa  =  2,720  +  1,020  =  3,740  Ibs.,  to  which  must 
be  added  the  weight  of  the  hub  to  obtain  the  total  weight  of  the 
wheel.  Since  the  rim  is  to  be  16  inches  wide  and  the  mean 

diameter  is  to  be  06  inches  the  thickness  will  be — 

o.26x7r  X96X  16 

=  2.2  inches  or  say  2%  inches. 

189.  Practical  Coefficients.  It  is  evident  that  the  ratio  of 
the  energy  to  be  absorbed,  to  the  total  energy  supplied  per 
energy  cycle,  will  vary  in  different  machines,  and  also  in  the 
same  machine  under  different  conditions.  Thus  in  the  punch- 
ing machine  the  flywheel  absorbs  and  redistributes  nearly  the 
entire  energy  supply  per  cycle,  while  in  the  steam-engine  example 
the  amount  absorbed  is  about  one-third  the  total.  It  is  readily 
seen  that  in  the  steam  engine  this  ratio  will  vary  with  the  point 
of  cut-off,  steam  pressure,  weight  of  reciprocating  parts,  etc.,  and 
therefore,  in  general,  tabulated  values  of  this  ratio  are  deceptive 
unless  they  refer  to  specific  conditions.  It  is  to  be  noted  that 
the  weight  of  the  flywheel  is  directly  proportional  to  the  energy  to 
be  absorbed  and  inversely  proportional  to  (v*  -  vj).  The  latter 


FLYWHEELS   AND   PULLEYS  413 

is  usually  a  small  quantity  and,  therefore,  if  E  is  large  the  weight 
of  the  flywheel  may  be  excessive,  which  is  undesirable  because 
of  the  cost,  and  also  because  heavy  wheels  bring  great  loads  on 
the  bearings,  causing  frictional  losses.  For  this  reason  it  is 
always  desirable  so  to  arrange  the  sequence  of  events  in  the  energy 
supply  and  work  to  be  done  as  to  minimize  the  excess  energy  to 
be  absorbed.  This  is  illustrated  in  Art.  6,  Fig.  5,  where  the  area 
K  may  be  greatly  decreased  (or  increased)  by  changing  the  relative 
positions  of  the  crank  pins.  This  procedure  is  of  great  import- 
ance to  avoid  wheels  of  great  weight  in  large  steam  engines  when 
variation  in  velocity  must  be  closely  restricted. 

The  allowable  variation  in  velocity  is  fixed  with  reference 
to  the  character  of  the  work  to  be  done.  It  is  evident  that  some 
classes  of  work  require  much  more  constant  velocity  than  others, 
and  experience  has  shown  what  the  limits  in  variation  of 
velocity  may  be  for  successful  operation.  The  following  limiting 

vi  —  ^2 
values  of  the  proportionate  variation  -  represent  average 

practice.  The  particular  case  of  direct  driving  of  alternating 
generators  in  parallel  must,  in  general,  be  treated  with  reference 
to  the  allowable  variation  per  pole,  and  when,  therefore,  the 
number  of  poles  is  great  the  total  allowable  variation  is  corre- 
spondingly small. 

TABLE  XXIX 


Values  of 


v 

For  punching  machines  and  similar  machines  ........  o.io    to  0.15 

For  engines  driving  stamps,  crushers,  etc  ...........  o  .  20 

For  engines  driving  pumps,  saw  mills    ..............  o  .03    to  o  .05 

For  engines  driving  machine  tools,  weaving  and  paper 

mills  .....................................  0-025  to  o  .03 

For  engines  driving  spinning  mills  for  coarse  thread  ....  o  .016  to  o  .025 

For  engines  driving  spinning  mills  for  fine  thread  .....  o.oi    to  0.02 

For  engines  driving  single  dynamos  .................  o  .007 

For  engines  driving  alternators  in  parallel    ..........  0.003  to  0.0003 

190.  Stresses  in  Flywheels.  The  velocity  of  the  rims  of  all 
flywheels  is,  from  the  nature  of  their  requirements,  very  high. 
If  the  wheel  is  to  act  as  a  band  wheel,  the  desirability  of  obtaining 


414  MACHINE   DESIGN 

high  belt  speed  (Art.  134)  brings  the  peripheral  velocity  up  to 
4,000  or  5,000  feet  per  minute.  It  has  been  shown  that  the 
capacity  of  a  given  wheel  is  proportional  to  the  square  of  its 
velocity  and,  therefore,  when  the  wheel  is  to  act  as  a  flywheel 
alone,  economy  in  the  use  of  material,  or  the  limiting  of  the 
external  dimensions,  makes  high  speed  very  desirable.  Great 
care  should  be  used  in  the  design  of  such  wheels,  for  a  flywheel 
which  breaks  at  normal  speed  is  exceedingly  dangerous  to  life  and 
limb,  and  when  such  wheels  "explode"  or  break  from  overspeed- 
ing,  the  results  are  usually  very  disastrous. 

Unfortunately,  mathematical  analysis  of  the  stresses  in  fly- 
wheels and  pulleys  is  not  satisfactory  or  conclusive.  In  the  case 
of  small  wheels  cast  in  one  piece,  unknown  shrinkage  stresses 
of  great  magnitude  may  exist,  which  render  useless  any  refined 
calculations.  In  large  wheels  built  up  of  sections,  the  presence 
of  joints  vitiates  any  calculations  based  on  the  elastic  theory  of 
the  strength  of  materials;  and  when  the  parts  are  of  cast  materials 
and  of  large  sectional  area,  there  is  no  assurance  that  the 
character  of  the  material  is  uniform  throughout.  It  is  impor- 
tant, however,  to  understand  fully  the  general  character  of  the 
stresses  even  when  no  accurate  computations  can  be  made  as  to 
their  magnitude. 

Consider  that  the  rim  of  the  wheel  in  Fig.  172  is  free  to  expand 
radially,  the  arms  exerting  no  restraining  force  in  a  radial  direc- 
tion. If  the  wheel  be  rotated  on  its  axis  the  action  of  centrifugal 
force  is  such  as  to  cause  an  outward  pressure  on  every  part  of  the 
rim,  in  exactly  the  same  manner  as  in  a  boiler  shell  acted  on  by 
an  internal  pressure  (see  Art.  78) ;  the  rim  expanding  until  the 
tensile  stress  induced  in  any  section  A  A,  Fig.  172,  balances  the 
tendency  of  the  wheel  to  separate  along  that  section.  If,  on  the 
other  hand,  the  arms  are  rigidly  attached  to  both  hub  and  rim, 
and  are  so  inelastic  that  their  stretch,  under  the  action  of  the 
centrifugal  pull  due  to  their  own  mass  and  that  of  the  rim,  is 
negligible,  it  is  clear  that  they  may  be  placed  so  close  together 
that  the  rim  cannot  exoand,  and  practically  no  stress  will  exist 
in  the  rim,  the  centrifugal  action  being  balanced  by  the  stress  in 
the  arms. 


FLYWHEELS   AND   PULLEYS 


415 


Flywheels  approximating  both  of  these  conditions  are  some- 
times built,  but  in  the  most  usual  case  the  arms  stretch  a  certain 
amount  and  are  not  placed  close  together,  so  that  a  condition  re- 
sults similar  to  that  shown,  in  an  exaggerated  manner,  in  Fig. 
173.  Here,  the  arms,  though  stretching  somewhat,  do  not 
stretch  enough  to  allow  the  rim  to  expand  freely,  and,  therefore, 
the  hoop  tension  is  somewhat  less  than  that  in  the  free  ring.  The 
section  of  rim  between  each  pair  of  arms  is  so  long  that  it  becomes 
a  beam  fixed  at  the  ends  and  loaded  uniformly  by  the  unbalanced 
centrifugal  action;  the  greatest  bending  moment  being  at  the  arm, 
and  a  bending  moment  of  half  the  maximum  occurring  at  the  centre 


FIG.  172, 


FIG.  173. 


of  the  span.  The  maximum  tensile  stress  will  be  the  sum  of  the 
hoop  tension  and  the  tensile  stress  due  to  the  bending  action. 
The  relative  values  of  the  hoop  tension  and  bending  stress  will, 
evidently,  depend  upon  the  amount  which  the  arms  stretch. 
If  they  should  stretch  enough  owing  to  their  own  centrifugal  force, 
so  that  the  rim  expands  freely,  no  bending  action  will  occur; 
while  if  they  are  so  inelastic  as  completely  to  restrain  the  rim, 
no  hoop  tension  will  be  induced,  but  the  full  centrifugal  force 
will  be  applied  to  bend  the  rim.  With  any  intermediate  amount 
of  stretch  of  the  arms  the  rim  will  be  held  in  equilibrium,  partly 
by  the  hoop  tension  and  partly  by  the  restraining  action  of  the 


41 6  •     MACHINE  DESIGN 

arms,  the  latter  being  a  measure  of  the  unbalanced  centrifugal 
force  of  the  rim,  and  of  the  bending  stress  caused  thereby.  Since 
the  expansion  of  the  rim  is  directly  proportional  to  the  stretch  of 
the  arms  it  is  clear  that  the  hoop  tension  is  also  directly  propor- 
tional to  the  stretch.  If,  for  instance,  the  arms  stretch  one 
quarter  the  amount  necessary  for  free  expansion,  the  hoop  tension 
will  be  one-quarter  that  due  to  free  expansion,  and  the  bending 
stress  will  be  proportional  to  three-quarters  of  the  centrifugal 
force  of  the  rim.  The  mathematical  relation  which  exists  between 
these  stresses  is  complex,  and  will  of  course  vary  with  the  relative 
size  and  shape  of  the  rim  arid  the  arms.  If  the  rim  is  of  a  wide 
thin  section,  and  the  arms  are  few,  the  bending  stress  may  be  very 
serious.  Professor  Lanza*  has  shown  that,  with  the  proportions 
ordinarily  used,  the  arm,  theoretically,  stretches  about  three- 
quarters  the  amount  necessary  for  free  expansion.  It  is  also  to 
be  noted  that  if  the  wheel  is  to  act  as  a  band  wheel,  and  has  a 
wide  thin  rim,  the  bending  action  on  the  arms  as  at  B,  Fig.  173, 
still  further  distorts  the  rim  and  increases  the  bending  on  the 
forward  'side. 

Let  D  =  the  mean  diameter  of  the  rim  in  feet. 

Let  R   =  the  mean  radius  of  the  rim  in  feet. 

Let  /     =  the  thickness  of  the  rim  in  inches. 

Let  v    =  the  velocity  of  the  rim  in  feet  per  second. 

Let  w  =  the  weight  of  the  material  per  cubic  inch. 

Let  /     =  the  length  of  the  rim  between  arms  in  inches. 

Consider  a  section  of  the  rim  one  inch  wide  on  the  face.  The 
centrifugal  force  per  unit  of  length  (i"),  circumferentially,  of 

wti? 
this  section  is  c  =  — —  and,  therefore,  by  Art.  78,  the  total  load 

wtv2 
which  tends  to  separate  such  a  ring  along  a  diameter  is  — — 

X  i2D,  and  the  unit  stress  in  the  section,  if  no  bending  exists, 
is  therefore, 

I2WtV*D          I21VV*          V* 

Pi  =   —£ =  — ,  nearly,  for  iron  wheels   .    (5) 

*  Trans.  A.  S.  M.  E.,  Vol.  XVI,  page  208. 


FLYWHEELS  AND   PULLEYS 


417 


The  maximum  bending  moment  in  the  rim  occurs  at  the 

cl2  * 

arms,    and    its    value    is   M  =  -     -  considering  the  rim  as  a 

12 

straight  beam.  The  stress  due  to  this  bending  moment  when  no 
hoop  tension  exists  is  therefore, 

_  M^.  _  cpe 

where  e  is  the  distance  to  the  outer  fibre,  and  /  j  the  moment  of 
inertia  of  the  cross-section  of  the  elementary  ring. 

If  now  the  stretch  of  the  arms  be  taken  as  three-quarters 
that  necessary  for  free  expansion  of  the  rim,  the  total  unit  tension  in 
the  rim  will  be 

/  o  T»2  r  T2  0\ 

-     •     .     (7) 

if  n  be  the  number  of  arms  in  the  wheel,  /  =  -       —   and  if  the 

n 

e       6 
cross-section  of  the  rim  be  rectangular  -  =  -^  whence  equation 

(7)  reduces  to 

'iv2       3  D  i?\  ( D  \ 

~  +  ~  r~2~7  =  3  v  \  —^  +  0.025 )     .     .     (8) 
40         tn*  '  Vw2  °/ 

For  t  =  — ,  v  =  88,  D  =  4  ft.,  and  n  =  6.     Professor  Lanza 

finds  the  stress  due  to  hoop  tension  =  575  and  the  stress  due  to 
bending  =  5, 060  or  p  the  total  stress  =  5,635.  For  the  same 
data  equation  (8)  gives  %A  =  581  and  %p2  =  4,600,  or  a  total 
stress  p  =  5,811,  which  agrees  quite  closely. 

The  above  equation  may  be  used  for  checking,  roughly,  the 
allowable  stress  in  flywheel  rims,  but  implicit  faith  must  not  be 
placed  upon  it  for  the  reasons  given  in  the  first  paragraph  of  this 
article,  and  all  results  obtained  from  this  or  similar  formulae 
should  be  checked  by  successful  practice  wherever  a  doubt  arises. 
The  equation  does,  however,  show  clearly  that  in  wheels  having 

*  See  Table  I,  case  17. 

fit  should  be  noted  that  this  I  is  for  a  unit  (i")  of  width  of  rim  and  not  for 
the  entire  cross-section. 
27 


4i 8  MACHINE  DESIGN 

thin  rims,  or  few  arms,  the  bending  stress  is  much  greater  than 
that  due  to  hoop  tension,  and  care  should  be  exercised  ac- 
cordingly when  such  wheels  must  run  at  high  speed.  Equation 
(5)  is  often  taken  as  a  basis  for  the  design  of  flywheels,  using, 
therewith,  a  large  factor  of  safety  to  cover  uncertainties.  If  pv 
in  equation  (5)  be  taken  as  1,000  (a  factor  of  safety  =  20),  then 
v  =  6,000  feet  per  minute,  and  this  is  found  to  be  a  safe  peripheral 
speed  for  ordinary  cast  iron  wheels.  It  is  to  be  noted,  however, 
that  this  speed  is  safe  only  because  experience  has  shown  it  to  be 
so,  and  not,  as  will  be  seen,  because  the  stress  is  necessarily  as 
low  as  1,000  pounds. 

Example.  Compute  the  stress  in  the  rim  of  the  cast-iron 
flywheel  discussed  in  example  (2)  of  Art.  188,  assuming  that  the 
arms  stretch  three-quarters  the  amount  necessary  for  free  expansion 

of  the  rim.     Here  n  =  6,  t  =  2.2,  D  =  8  and  v  =  -      -  =  66.6. 

oo 

.-.  from  (8),  p  =  3  v2  (— 2  +  0.025)  = 

3  X  66. 62  ( —2  +  0.025)  =  1,668  Ibs.  per-sq.  in. 

The  stress,  if  based  on  equation  (5),  would  be  444  pounds  per 
square  inch. 

When  a  flywheel  is  being  accelerated  from  rest,  or  when  the 
energy  supply  is  suddenly  cut  off,  as  it  may  be  in  a  steam  engine, 
the  arms  may  be  called  upon  to  carry  the  full  torque  load.  Each 
arm  of  a  wheel  with  a  very  stiff  rim  approximates  a  cantilever 
beam  fixed  at  one  end,  free  but  guided  at  the  other,  and  car- 
rying a  concentrated  load  at  the  free  end  (see  Table  I,  case  7) . 
If  the  rim  is  thin  and  flexible,  the  arms  approximate  a  simple 
cantilever  loaded  at  the  free  end.  In  addition,  the  arm  is  sub- 
jected to  a  tensile  stress  due  to  the  centrifugal  action  of  its  own 
weight,  and  that  part  of  the  rim  which  it  supports,  so  that  appar- 
ently equation  M  (Table  VI)  applies.  The  direct  stress  is 
difficult  to  compute,  however,  and  since  the  bending  stress  in 
the  simple  cantilever  is  twice  that  of  a  cantilever  with  the  free  end 


FLYWHEELS   AND   PULLEYS  419 

guided,  it  is  considered  sufficiently  accurate  to  compute  the  arm 
as  a  simple  cantilever  and  neglect  the  direct  stress. 

Let  P  =  the  greatest  force  due  to  the  belt  pull  at  the  rim. 

Let  a  =  the  length  of  the  arm. 

Let  n  =  the  number  of  arms. 
Then  from  J  (Table  VI)  : 

Pae 


from  which  the  stress  p,  or  the  moment  of  inertia  7,  may  be 
determined.  The  stress  allowed  should  not  exceed  2,000  pounds 
per  square  inch,  for  cast  iron,  on  account  of  the  uncertainties 
of  the  case,  and  a  lower  value  is  sometimes  desirable.  The 
statement  sometimes  made  that  the  arms  should  be  as  strong 
against  bending  stress  as  the  shaft  is  against  torsional  stress,  is 
misleading  as,  in  general,  shafts  are  designed  for  stiffness,  and 
not  for  torsional  strength.  The  shaft  of  a  steam  engine  may 
have  to  be  very  large  to  avoid  excessive  deflection  and,  as  a 
consequence,  may  have  great  excess  of  torsional  strength. 

191.  Construction  of  Wheels.  Flywheels  and  band  wheels, 
for  velocities  below  5,000  feet  per  minute,  are  usually  made  of 
cast  iron  on  account  of  low  cost.  For  higher  velocities  steel 
castings  are  used,  and  in  extreme  cases  wheels  made  of  steel 
plates,  or  wire-  wound  wheels  have  been  constructed.  ,  Equation 

(5)  may  be  written  v  =  1.64  x  A     The  allowable   unit   tensile 

^  w 

strength  divided  by  the  weight  per  cubic  unit  is,  therefore,  a 
measure  of  the  value  of  the  material  for  this  purpose.  For  this 
reason  some  woods  are  superior  to  cast  iron  for  wheel  rims,  and 
cast-iron  wheels  which  have  burst  have  been  successfully  re- 
placed with  wheels  having  rims  made  of  wood.* 

Difficulties  in  transportation  limit  the  diameter  of  wheels 
cast  in  one  piece  to  about  ten  feet,  and  the  diameter  of  wheels 
.cast  in  two  parts  to  about  twenty  feet.  Wheels  from  about  six- 
teen feet  in  diameter  upward  are  usually  made  in  several  sections. 
Small  flywheels  and  band  wheels  are  usually  cast  in  one  piece,  or 

*  Trans.  A.  S.  M.  E.,  Vol.  XIII,  page  618. 


420 


MACHINE  DESIGN 


made  in  two  parts  for  convenience  in  erecting.  In  either  of  the 
latter  cases  unknown  shrinkage  stresses  will  most  probably  exist. 
These  shrinkage  stresses  are  sometimes  relieved  by  casting  the 
hub  in  several  pieces,  each  piece  being  cast  integral  with  one  or 
more  arms.  The  openings  between  the  parts  are  afterward  filled 
with  lead,  and  rings  are  shrunk  upon  the  hub  to  hold  the  parts 
in  place.  Experience  shows  that  solid  cast-iron  wheels,  when 
properly  proportioned,  are  safe  up  to  6,000  feet  per  minute  which, 
fortunately,  is  also  about  the  limit  of  efficient  belt  speed.  If,  how- 
ever, the  wheel  has  a  very  wide  thin  rim  it  cannot  be  considered 
safe  at  this  speed,  particularly  if  balance  weights  are  attached  to 
the  rim  between  the  arms,  thus  increasing  the  centrifugal  bending 


FIG.  174. 


FIG.  175. 


force.  If  joints  exist  in  the  rim,  their  relative  strength  must  be 
considered.  Band  wheels  of  wrought-steel  construction  can  now 
be  obtained  up  to  about  4  feet  in  diameter;  they  are  light  and 
strong,  and  are  rapidly  coming  into  favor. 

Where  speeds  above  6,000  feet  per  minute  are  necessary, 
wheels  such  as  shown  in  Fig.  174  are  sometimes  built.  Here 
the  rim  and  hub  are  of  cast  iron,  each  cast  in  one  piece,  and  the 
spokes  are  of  steel.  The  spokes  are  placed  in  the  mould,  and 
the  metal  poured  around  them,  so  that  on  cooling  they  are  gripped 
very  firmly.  The  spokes  are  placed  close  together  so  that  there 
is  practically  no  bending  of  the  rim,  and  the  rim  is  also  prevented 
from  expanding  freely.  Wheels  of  this  construction  are  used  for 
large  band  saws  at  velocities  above  10,000  feet  per  minute, 
under  heavy  service,  with  perfect  success. 


FLYWHEELS   AND   PULLEYS 


421 


In  Fig.  175  the  rim  is  cast  separately  in  one  or  more  pieces. 
The  arms  do  not  constrain  the  rim  radially,  but  leave  it  free  to 
expand.  The  stresses  in  the  rim  when  cast  in  several  pieces  so 
that  shrinkage  is  not  a  factor  are  those  due  to  centrifugal  force 
only,  and  the  arms  are  simple  cantilevers.  Wheels  of  this  charac- 
ter have  been  used  with  success  in  rolling-mill  work.*  Figs.  174 
and  175  illustrate  wheels  which  correspond  closely  to  the  limiting 
types  discussed  in  Art.  190.  The  construction  of  most  wheels 
lies  between  these  types.  Fig.  176  illustrates  a  band  wheel  with 
the  arms  and  hub  cast  in  one  piece  and  the  rim  in  sections.  The 
joints  in  the  rim  are  simple  flange  joints,  placed  midway  between 
the  arms.  This  is  the  most  dangerous  location  possible,  on 
account  of  the  added  bending  effect  due  to  the  centrifugal  force 


FIG.  176. 


FIG.  177. 


of  the  flanges  which  add  to  the  mass  without  contributing  to  the 
strength.  The  best  location  is  at  the  arm,  and  many  wheels  are 
built  thus,  the  arm  being  bolted  to  each  segment,  and  the  seg- 
ments themselves  bolted  together  as  well.  Where  the  joint  is 
placed  between  the  arms,  it  should  be  about  one-quarter  the 
•length  of  the  span  away  from  the  arm,  as  at  A,  Fig.  176,  where, 
by  the  theory  of  elasticity,  the  bending  moment  is  zero.  Fig. 
177  shows  a  heavy  flywheel  with  an  arm  and  a  segment  of  the 
rim  cast  together.  The  arms  are  secured  in  the  hub  by  means 
of  fitted  bolts.  The  hub  may  be  solid  or  the  flange  on  one  side 
may  be  movable  axially  so  as  more  firmly  to  clamp  the 
arms.  The  segments  are  held  together  at  the  rim  by  means 
of  links  of  rectangular  cross-section  shrunk  in  place.  This 
construction  is  very  common.  Occasionally  links  are  also 

*  Trans.  A.  S.  M.  E.,  Vol.  XX,  page  944. 


422 


MACHINE  DESIGN 


shrunk  into  recesses  on  the  outer  face  of  the  wheel.  In  Fig.  178 
the  segments  are  held  together  by  T-headed  links,  sometimes 
called  "prisoners,"  shrunk  in  place.  The  segments  are  joined 
at  the  arms,  which  are  fastened  to  them  by  through  bolts. 
This  construction  is  simple  and  the  machining  is  easier  than  with 
flanged  connections.  The  construction  of  the  hub  is  similar  to 
that  in  Fig.  177. 


FIG.  178. 


FIG.  179. 


It  is  evident  that  the  manner  of  joining  segments  in  built-up 
wheels  is  most  important.  Wheels  seldom  fail  at  the  hub. 
Wheels  with  thin,  wide  sections  are  almost  always  joined  by 
flanges  as  shown  in  Fig.  180.  When  such  joints  are  used  they 
should  be  well  ribbed  for  stiffness,  as  indicated,  and  the  bolts 
should  be  placed  as  near  the  rim  as  possible,  so  that  the  lever 
arm  a  shall  be  as  great  as  possible  compared  to  the  arm  b  (see 


FIG.  180. 


FIG.  181. 


FIG.  182. 


Art.  63).  A  much  better  arrangement  is  shown  in  Fig.  181, 
where  an  arm  is  placed  on  each  side  of  the  joint.  This  is  par- 
ticularly applicable  to  wheels  cast  in  two  parts.  It  may  be  noted 
that  thin  rims  are  often  stiffened  by  light  circumferential  ribs  at 
the  outer  edges.  Mr.  A.  K.  Mansfield  has  pointed  out  (Trans. 
A.  S.  M.  E.,  Vol.  XX)  that  these  ribs  maybe  a  source  of  weakness. 


FLYWHEELS  AND  PULLEYS  423 

The  greatest  bending  moment  is  near  the  arm  where  these  ribs 
are  on  the  tension  side  of  the  beam.  A  rim  having  such  ribs  is 
not  necessarily  as  strong  against  bending  in  this  direction,  as  one 
of  rectangular  cross-section  having  the  same  area  ;  and  when 
ribs  are  used  the  section  modulus  should  be  calculated. 

The  prisoner  link  shown  in  Fig.  178  has  certain  advantages  over 
the  link  shown  in  Fig.  177.  It  is  evident  that  the  depth  of  the 
recesses  in  Fig.  177  is  limited,  while  in  Fig.  178  the  slot  can  extend 
entirely  across  the  section  and  the  link  can  be  made  as  wide  as 
the  rim  itself.  Furthermore,  it  is  possible  to  machine  both  wheel 
and  link  in  Fig.  178  accurately,  which  is  difficult  to  do  with  the 
construction  in  Fig.  177.  This  permits  of  greater  accuracy  in 
computing  the  initial  stress  induced  in  the  link  by  shrinking  it 
in  place,  the  importance  of  which  has  been  noted  in  Art.  77.  If 
the  rim  be  made  I-shaped,*  as  in  Fig.  182,  the  links  can  be  so  pro- 
portioned that  the  joint  will  be  as  strong  as  the  rim  proper  or 
even  stronger. 

While,  evidently,  the  relative  strength  of  the  joint  com- 
pared to  the  solid  rim  will  vary  with  the  exact  proportions  selected, 
average  practice  gives  the  following  apparent  values: 

Flanged  joint,  bolted,  midway  between  arms    25 

Flanged  joint,  bolted,  at  end  of  arms    50 

Linked  joint  as  in  Fig.  177     60 

Linked  joint  as  in  Fig.  178     65 

Linked  joint  as  in  Fig.  182     i  .00 

Solid  rim   i .  oo 

It  must  not  be  inferred  from  the  above  that  a  solid 
rim  is  necessarily  the  best;  as,  obviously,  a  wide  thin  rim 
with  unknown  shrinkage  strain  may  not  be  as  safe  as  a  narrow 
deep  rim  of  the  same  sectional  area  if  held  together  by  a  good 
joint. 

For  extreme  velocities,  wheels  built  up  of  steel  plates,  or 
wheels  with  rims  made  of  plates  fastened  to  a  central  spider  made 
of  steel  castings,  are  now  used.  Fig.  179  shows  a  flywheel  of  the 
latter  type  used  in  rolling-mill  work  (see  Power,  Feb.  4,  1908). 
The  rim  is  made  of  laminations  held  to  the  spider  by  dovetails, 

*  See  Trans.  A.  S.  M.  E.,  Vol.  XX,  page  944- 


424  MACHINE   DESIGN 

as  shown,  the  laminations  being  assembled  with  overlapping 
joints.  Heavy  outside  plates  clamp  the  whole  structure  together 
by  means  of  through  bolts.  In  the  particular  case  noted  above, 
the  velocity  of  the  wheel  rim  is  250  feet  per  second.  Descriptions 
of  a  number  of.  examples  of  such  wheels  are  to  be  found  in  the 
Transactions  of  the  A.  S.  M.  E.,  the  magazine  Power,  and  other 
periodicals.  Wheels  for  great  speed  have  also  been  constructed 
by  winding  the  rim  with  many  turns  of  steel  wire. 

The  rotors  of  some  forms  of  electric  generators,  steam  turbine 
rotors,  and  similar  revolving  members  are  often  loaded  as  shown 
at  W,  Fig.  176.  Such  loads  add  to  the  centrifugal  force  acting 
on  the  rim,  but  do  not  add  to  the  strength  of  the  rim.  Due  al- 
lowance should  be  made  in  such  cases;  particularly  if  the  load 
or  loads  are  placed  near  a  joint  as  shown  in  Fig.  176.  The 
teeth  of  gear  wheels  constitute  such  a  load,  and  if  the  wheel  is  large, 
and  the  peripheral  speed  high,  this  should  be  considered.  Balance 
weights,  placed  between  the  arms,  should  be  carefully  considered, 
especially  when  the  rim  is  thin  and  the  velocity  high. 

192.  Experiments  on  Flywheels.  The  best  experimental  data 
upon  the  strength  of  flywheels  are  from  tests  conducted  by 
Professor  Benjamin  and  reported  to  the  A.  S.  M.  E.*  While 
these  experiments  were  made  to  determine  the  bursting  speed  of 
small  cast-iron  wheels  only,  and  throw  no  light  on  the  increase  of 
stress  with  an  increase  of  speed,  they  are  very  valuable  as  indicat- 
ing the  manner  in  which  various  types  of  wheels  fail.  Being 
conducted  on  small  wheels,  due  allowance  must  be  made  for  the 
difference  in  quality  between  the  metal  of  small  and  large  castings 
in  estimating  probable  bursting  stresses.  These  experiments  go 
to  show  that  solid  cast  wheels  will  burst  at  a  peripheral  velocity 
somewhere  near  400  feet  per  second,  and  such  wheels  are  safe  only 
at  a  velocity  of  not  more  than  100  feet  per  second.  Rim  joints 
midway  between  the  arms,  particularly  the  common  flange  joints, 
were  found  to  reduce  the  strength  materially.  The  strength 
of  various  joints  was  found  to  be  about  as  tabulated  in  Art.  191. 

*  See  Trans.,  Vols.  XX  and  XXIII.  See  also  "Machine  Design,"  by  C,  H. 
Benjamin. 


FLYWHEELS  AND  PULLEYS  425 

Extra  loads,  such  as  balance  weights  located  between  the  arms, 
were  found  to  be  very  dangerous,  on  account  of  the  added  bending 
effect. 

193.  Rotating  Discs.  If  the  radial  depth  of  a  wheel  rim  be 
great  compared  to  its  axial  width,  the  equations  deduced  in  the 
preceding  articles  do  not  apply,  the  difference  being  analogous 
to  that  existing  between  thick  and  thin  cylinders.  Mathematical 
analysis  of  the  stresses  in  a  rotating  disc,  in  common  with  those 
existing  in  thick  cylinders  under  internal  pressure,  are  complicated 
and  not  altogether  satisfactory.  Experimental  data,  corroborat- 
ing the  theories,  are  also  lacking.  A  full  mathematical  treatment 
of  these  stresses  is  beyond  the  scope  of  this  treatise,  and  only 
enough  will  be  inserted  to  show  the  general  character  of  the 
problem. 

When  a  disc  of  uniform  thickness  is  rapidly  rotated  on  its 
axis,  the  principal  stresses  induced  are  a  tangential  tension,  and 
a  radial  stress,  at  every  point  in  the  disc. 

Let  r2  =  the  outer  radius  of  the  disc  in  inches. 

"  r^  =  the  inner  radius  of  the  disc  in  inches. 

"  r    =  the  radius  at  any  point. 

"  ^    =  Poisson's  ratio  =  yz  for  steel  and  %  for  cast  iron. 

"  N  •=  revolutions  per  minute. 

"   w  =  weight  of  one  cubic  inch  of  the  material. 

"   p   =  the  tangential  stress  at  any  radius  r. 

"   p*  =  the  radial  stress  at  any  radius  r. 
Then  it  can  be  shown*  that  for  a  flat  disc  of  uniform  thickness, 
having  a  hole  at  the  centre  of  radius  rly- 


[ 


(n) 


and  f  =  o.  00000355  wN* 

For  a  solid  disc 

p  =  o.  00000355  w  N2  [  (3  +  A)  r22  -  (i  +  3 
and  /  =  0.00000355  wN2  [  (3  +  /)  (r22  -  r2)  ]   .     (14) 

*  See  "Theory  of  the  Steam  Turbine,"  by  A.  Jude,  pages  192  and  204.     The 
notation  and  units  have  been  changed  to  correspond  with  those  used  in  this  text. 


426  MACHINE   DESIGN 

It  is  to  be  noted  that  the  radial  stress  is  less  at  any  point  than 
the  corresponding  tangential  stress;  and  an  examination  of 
equation  (n)  shows  that  this  tangential  stress  is  a  maximum  at 
the  surface  of  the  bore  and  a  minimum  at  the  outer  periphery. 
At  the  surface  of  the  bore  or  where  r  =  r,  the  stress 

p  =  0.00000355  a  N2  [  (3  +  A)  (2  r22  +  r,2)  -  (i  +  3  A)  r,2] 

If  now  rl  be  taken  so  small  that  rt2  is  negligible,  it  appears 
that  the  tangential  stress  is 

p  =  2  X  0.00000355  «  N2  [  (3  +  A)  /22] 

which  is  just  twice  that  obtained  by  making  r  =  o  in  equation 
(13).  The  effect  of  even  a  very  small  hole  at  the  centre  of  a 
rotating  disc  is,  therefore,  to  increase  the  stresses  greatly. 

Example.  A  circular  steel  saw  %  inch  in  thickness  and  80 
inches  in  diameter  has  a  hole  4  inches  in  diameter  in  the  centre 
and  runs  at  the  rate  of  500  R.P.M.  Determine  the  tangential 
stress  at  rim  and  also  at  the  hole. 

Here  N  =  500,  w  =  0.28,  ^  =  K,  r2  =  40,  and  r,  =  2. 
Whence  in  (n)  making  r  =  r2  =  40  the  tangential  stress  at  the 
rim  is 

p  =  0.00000355  X  0.28  X  5oo2  [  (3  +  X)  (4o2  +  22  +  22)  - 
(i  +  i)  4o2]  =  535  Ibs.  per  sq.  in. 

and  at  the  hole  making  r  =  rl  =  2. 

p  =  0.00000355  X  0.28  X  5oo2  [  (3  +  K)  (402  +  22  +  4o2)  - 

(i    +    i)  22]  =   2,643   Iks.   Per  SCl-  m' 

The  foregoing  equations,  (u)  to  (14),  are  deduced  on  the 
hypothesis  that  the  material  is  perfectly  elastic  and  homogeneous. 
It  is  clear  that  they  cannot  be  intelligently  applied  to  built-up 
wheels  of  the  disc  type,  and  must  also  be  applied  with  caution  to 
brittle  materials.  They  are  of  great  value,  however,  in  showing 
the  general  character  of  these  stresses  and  the  location  of  the 
greatest  stress,  thus  indicating  the  shape  which  discs  should  have 
for  uniform  strength;  for  a  brief  reflection  will  show  that  such 
discs  must  be  thickened  at  the  centre  to  reduce  the  stress  at  that 


FLYWHEELS  AND   PULLEYS  427 

point.  For  complete  mathematical  analysis  of  discs  of  different 
shapes,  reference  may  be  made  to  the  various  works  on  the  steam 
turbine.  It  is  evident  that  great  care  should  be  used  in  selecting 
and  working  the  material  for  high-speed  discs.  Rolled  sheets 
are  not  good  for  very  high  speeds  on  account  of  their  seamy 
structure,  which  is  conducive  to  incipient  cracks,  and  cast  ma- 
terials of  brittle  structure  must  be  of  first-class  quality.  Discs 
forged  down  from  much  thicker  ingots  give  the  safest  construction. 

References : 

"The  Theory  of  the  Steam  Turbine,"  by  A.  Jude. 

"Steam  Turbines,"  by  L.  French. 

"The  Steam  Turbine,"  by  Dr.  A.  Stodola. 


CHAPTER  XVI 
MACHINE  FRAMES  AND  ATTACHMENTS 

194.  Stresses  in  Machine  Frames.  Since  machine  frames 
must,  in  general,  receive  the  reactions  from  the  forces  applied 
to  the  various  moving  members  by  the  energy  transmitted,  it  is 
obvious  that  the  stresses  induced  in  frame  members  are,  in  most 
cases,  very  complex  and  beyond  mathematical  analysis.  If  it  is 
essential  that  the  moving  members  be  held  in  accurate  alignment, 
as  in  the  case  of  machine  tools,  the  predominating  requirement 
for  the  frame  is  stiffness  and  not  strength.  For  these  reasons 
the  design  of  machine  frames,  in  general,  must  be  governed  largely 
by  judgment  and  experience,  the  cases  where  complete  mathe- 
matical analysis  is  possible  being  rare.  However,  even  in  cases 
where  judgment  must  be  the  guide,  it  is  not  only  helpful,  but 
sometimes  necessary  to  check,  as  closely  as  possible,  the  stresses 
in  certain  important  sections,  by  applying  those  fundamental 
formulae  of  Table  VI,  page  94,  which  apparently  fit  the  circum- 
stances. In  all  cases,  what  may  be  termed  a  ^qualitative 
analysis"  of  the  frame  is  very  desirable  as  a  guide  in  properly 
distributing  the  material,  and  in  determining  the  forms  of  the 
various  sections. 

If  the  character,  value,  and  line  of  action  of  every  force  acting 
upon  a  given  section  are  known,  the  stresses  in  the  section  can  be 
determined  by  applying  the  fundamental  requirements  for  static 
equilibrium  of  the  section,  namely  :— 

(a)  The  algebraic    sum   of   all   horizontal  component  forces 

must  =  o. 

(b)  The  algebraic  sum  of  all  vertical  component  forces  must 

=  o. 

(c)  The  algebraic  sum  of  all  the  moments  must  =  o 

428 


MACHINE   FRAMES   AND  ATTACHMENTS 


429 


The  stress,  in  any  direction,  at  any  point,  will  be  the  algebraic 
sum  of  all  the  stresses  acting  in  that  direction,  at  that  point,  as 
found  by  applying  (a),  (b),  and  (c).  It  is  impossible  to  make  a 
classification  of  machine  frames  that  would  be  of  any  service,  but 
the  principles  will  be  illustrated  by  applying  them  to  typical  cases. 
It  is  to  be  noted  that  it  is  seldom  possible  to  find  the  required 
dimensions  of  a  section,  directly,  by  solving  the  particular  equa- 
tions from  Table  VI  which  apply;  but,  in  general,  the  section 
must  be  assumed  from  the  conditions  given,  and  then  checked 
for  strength  or  stiffness. 

Fig.  183  illustrates  a  type  of  frame  which  is  quite  common 
and  known  as  an  open  frame.  It  is  one  of  the  few  types  where 


FIG.  183. 


FIG.  184. 


a  mathematical  analysis  can  be  made  with  some  degree  of  com- 
pleteness. In  the  case  of  a  punching-machine  frame  as  illustrated 
in  Fig.  183,  great  stiffness  is  not  essential  and  the  design  may 
be  based  on  the  strength  required.  Suppose  the  frame  to  be 
outlined  as  shown  so  that  the  dimensions  of  the  cross-section  at 
any  place  maybe  assigned.  Evidently,  if  the  stresses  are  checked 
at  the  sections  BC,  DE,  and  FG,  the  strength  of  the  frame  will  be 
fully  determined. 

In  the  section  BC,  whose  gravity  axis  is  at  Olt  consider  the 
portion  of  the  frame  above  BC  as  a  free  body.  It  is  in  equilibrium 
under  the  action  of  the  exterior  force  P,  due  to  punching,  and  the 


430  MACHINE   DESIGN 

internal  forces  exerted  upon  it  by  the  lower  half  of  the  frame. 
There  are  no  horizontal  forces.  The  vertical  force  P  must  be 
balanced  by  an  equal  and  opposite  force  at  the  section  BC,  which 
induces  a  tensile  stress  uniformly  distributed  over  the  section, 

p 

the  intensity  of  which  is  pl  =  —  pounds  per  square  inch,  where 

A  is  the  area  of  the  section.  The  only  moment  acting  on  the  part 
is  Pa,  due  to  the  action  of  P,  which  tends  to  rotate  the  upper  part 
of  the  frame  around  Olt  the  gravity  axis  of  the  section,  causing  a 
resisting  tension  at  B,  and  a  resisting  compression  at  C.  The 
maximum  intensity  of  these  flexural  stresses  is  given  by  the 
fundamental  equation  for  flexure  in  beams  (see  J,  Table  VI),  or 

Pae 

p2  =  —j —  where  e  is  the  distance  from  Ox  to  the  outer  fibre 
A 

and  II  is  the  moment  of  inertia  of  the  cross-section  around  the 
axis  Oi.  The  greatest  tension  will  therefore  be  at  B  and  its  value 

will  be  p  =  pi  +  p2  =  —  +  -     — ,  which   is    equation    M   of 
A.  e 

Table  VI.  This  is,  therefore,  a  case  of  combined  flexure  and 
direct  stress,  which  is  fully  discussed  in  Art.  19,  Chapter  III. 

Consider  next  the  section  DE,  whose  gravity  axis  is  at  O2, 
and  suppose  the  part  of  the  frame  at  the  left  of  DE  to  be  a  free 
body.  There  are  no  horizontal  forces  and  the  vertical  force  P 
must  be  balanced  by  a  vertical  tensile  pull  upon  the  upper  part 
of  the  frame  by  the  lower  part.  The  resultant  of  this  tensile 
pull,  which  is  distributed  uniformly  over  the  whole  section,  may 
be  represented  by  O2  K  acting  at  the  centre  of  gravity.  This 
force  may  be  resolved  into  the  components  HK  =  P1  perpen- 
dicular to  DE,  and  producing  a  tensile  stress  at  right  angles  to 
the  section  and  P2  =  O2  H  parallel  to  DE  and  producing  a 
shearing  stress  along  the  section.  The  only  moment  acting  upon 
the  section  is  that  due,1  as  before,  to  P,  whose  moment  arm  is  a2. 
The  tensile  and  compression  stress  due  to  this  moment,  as  deter- 
mined by  equation  J,  Table  VI,  may  be  combined  with  the  direct 
stress  PJ,  as  in  the  section  BC,  to  find  the  maximum  tensile  or 
compression  stress.  The  shearing  stress  is 


MACHINE  FRAMES  AND  ATTACHMENTS  431 

p 
ps  =  — -  where  A2  =  the  area  of  the  section  DE. 

A2 

This  is  usually  small  and  may  be  neglected  except  near  the 
ends  of  the  beam  as  in  the  section  FG  (see  Art.  14,  Chapter  III). 

Consider  last  the  section  FG.  As  before,  there  are  no  hori- 
zontal forces,  but  the  vertical  force  P  must  be  balanced  by  a 
vertical  resisting  force  which  induces  a  shearing  stress  at  the 

p 

section.      The  intensity  of  this  shearing  stress  is  — ,  where  A3 

A* 

is  the  area  of  the  section.  Since  the  area  of  the  section  is  much 
smaller  than  at  DE  or  BC,  it  is  advisable  to  compute  its  value. 
The  moment  Pa3  is  balanced  as  before  by  the  resisting  moment 
of  the  section  and  the  resulting  stress  may  be  computed  by  equa- 
tion 7,  of  Table  VI.  Evidently  these  general  principles  may  be 
applied  to  any  section. 

Fig.  185  illustrates  an  open  frame  as  applied  to  a  power 
riveter.  The  rivet  which  is  to  be  " driven"  is  placed  between  the 
dies  D  and  Dv  and  pressure  is  applied  to  the  movable  die  Z),  by 
means  of  the  power  cylinder  R.  The  pressure  which  is  applied 
may  be  very  great  (150  tons  or  more),  and  unless  the  jaws  are 
properly  designed  they  may  spring  so  much  that  the  dies  will 
fail  to  align  properly,  and  faulty  work  will  result  (see  Art.  53). 
Stiffness  and  not  strength  is,  therefore,  the  essential  factor  in 
the  design;  for  if  the  parts  are  stiff  enough  they  will  be,  in 
general,  strong  enough.  The  yielding  which  most  affects  the 
alignment  is  that  due  to  the  bending  of  the  frame  B,  and  the 
stake  C,  and  that  which  may  result  from  the  elongation  of  the  bolts 
which  hold  these  members  together.  When  the  riveting  pressure 
P  is  applied,  the  beams  B  and  C  tend  to  rotate  around  the  point 
O,  this  tendency  being  resisted  by  the  tension  in  the  bolts.  The 
load  which  may  be  applied  to  the  bolts  by  the  force  P  will  be 

Pl  =  -    — .     If  the  nuts  on  the  bolts  are  set  up  so  that  a 

combined  total  initial  tension  somewhat  greater  than  Pl  is  induced 
in  the  bolts,  the  stretching  of  the  bolts,  and  the  consequent  open- 
ing up  between  the  frame  and  the  stake,  will  be  negligible. 


432 


MACHINE  DESIGN 


(See  Art.  60  and  Fig.  43  and  also  Art.  77.)  The  intensity  of 
stress  in  the  bolts  should  not  exceed  6,000  pounds  per  square  inch. 
The  upper  part  of  the  frame,  B,  approximates  a  cantilever  of 
uniform  strength  of  length  a.  (See  Art.  15  and  Case  i  of  Table 
II.)  The  maximum  deflection  which  occurs  at  D  may,  therefore, 
be  computed  and  the  maximum  stress  which  occurs  at  E  F  may 
be  checked  by  Equation  J  of  Table  VI.  The  stake,  C,  approxi- 
mates a  cantilever  of  uniform  cross-section,  and  may  therefore  be 


FIG.  186. 


treated  in  a  similar  manner.  (See  Case  i,  Table  I,  and  Equation 
J,  Table  VI.) 

Fig.  1 86  illustrates  a  closed  frame  as  applied  to  a  vertical 
steam  engine.  The  back  column,  B,  which  carries  the  crosshead 
guide  is  of  cast  iron,  while  the  front  columns,  C,  are  of  steel.  It 
is  required  to  check  the  stresses  in  these  columns  when  the  piston 
is  ascending  and  also  when  it  is  descending,  the  rotation  of  the 
engine  to  be  taken  in  a  clockwise  direction  as  indicated. 

When  the  piston  is  ascending,  the  steam  pressure  tends  to 
draw  the  cylinder  and  bed  closer  together.  This  tendency  is 
resisted  by  P',  the  combined  thrust  on  all  the  columns,  the  vertical 
component  of  which  must  equal  P,  the  total  steam  pressure  on  the 
piston.  It  may  be  reasonably  assumed  that  the  back  column 


MACHINE  FRAMES  AND  ATTACHMENTS  433 

carries  one-half  of  the  total  thrust,  and  that  each  of  the  front 

P' 

columns  carries  one-quarter.     The  thrust  of  the  back  column, — , 

2 

may  be  resolved  into  components  perpendicular  and  parallel  to 

p 

the  face  of  the  foot.     The  vertical  component  will  equal  — .     The 

horizontal  component  R  tends  to  spread  the  foot  of  the  column 
outward  and  induce  a  bending  stress  in  it.  The  column  should, 
therefore,  be  secured  to  the  bed  by  fitted  bolts,  or,  if  the  bolts 
are  loose  in  the  holes,  the  foot  should  be  well  dowelled  to  the  bed; 
or,  better  still,  the  foot  should  fit  against  a  ledge  cast  on  the  bed 
plate.  R  will  then  be  balanced  by  an  equal  and  opposite  reaction 
at  the  feet  of  the  front  columns,  thus  setting  up  a  negligible  tension 
in  the  bed  and  leaving  a  compressive  force  only  on  the  column. 
By  similar  reasoning  each  front  column  is  subjected  to  a  com- 

Pf 
pressive  load  —  and  the  total  horizontal  component  R  is  balanced 

by  that  of  the  back  column  through  the  bed. 

The  tension  or  compression  in  the  piston-rod  and  connecting- 
rod,  either  ascending  or  descending,  have  a  resultant  R'  normal 
to  the  guide,  which  may  have  a  large  value  where  the  connecting- 
rod  is  short  compared  to  the  crank.  This  resultant  tends  to 
bend  B,  and  hence  C  also,  in  a  left-hand  direction,  the  bending 
being  resisted  by  the  fastenings  at  the  feet.  The  columns  and 
cylinder,  however,  constitute  a  very  stiff  structure,  and  except 
where  the  frame  is  made  up  of  light  construction  this  effect  may 
be  neglected.  This  reaction,  R',  however,  also  bends  the  column 
B  locally,  that  is  as  a  beam  encastre  at  S  and  N,  the  effect  of  R 
being  greatest  when  the  crosshead  is  near  half  stroke.  (See 
Case  1 8,  Table  I.)  If  then  it  be  desired  to  check  the  central 

P' 
section   UV  of  the  column,  the  long  column  stress  due  to  — -  must 

be  added  to  the  flexural  stress  due  to  R'.  The  sum  of  these 
stresses  should  not,  of  course  be  greater  than  the  allowable  stress 
for  the  material  used.  The  columns,  C,  need  only  be  checked 
as  long  columns  (see  equation  N,  Table  VI) . 


434  MACHINE   DESIGN 

When  the  piston  is  descending  the  steam  pressure  tends  to 
separate  the  bed  and  the  cylinder.  The  reactions  at  M  and  N 
are  reversed  in  direction  and  the  columns  are  put  in  tension,  the 
horizontal  components  inducing  negligible  compression  in  the 
bed.  The  most  dangerous  section  in  this  case  will  be  under  R' 

Pr 
and  the  stress  will  be  that  due  to  —  plus  the  tensile  stress  due  to 

the  bending  effect  of  R' '.  The  fastenings  of  the  columns  to  the 
cylinder  and  to  the  bed  plate  must,  of  course,  be  sufficiently  strong 
in  tension  to  resist  the  force  tending  to  separate  the  cylinder  and 
bed. 

In  the  foregoing  examples  the  lines  of  action  of  all  forces 
acting  on  the  section  considered,  lay  in  a  plane  of  symmetry  of 
the  section,  and  the  section  tended  to  rotate  around  a  gravity 
axis  at  right  angles  to  this  plane.  While  this  is  the  most  usual 
case,  occasionally  the  force  or  forces  acting  are  not  in  a  plane  of 
symmetry.  Thus  Fig.  187  may  represent  the  cross-section  of  the 
column  of  a  radial  drilling  machine,  in  which  it  is  required  to 
check  the  stresses  when  the  force  P,  due  to  drilling,  is  in  the 
position  shown.  If  C  be  the  centre  of  gravity  of  the  section,  the 
tendency  to  rotate  will  be  around  the  axis  X'  X'  at  right  angles  to 
PC,  the  arm  of  the  force  P,  and  the  resistance  of  the  section  against 
such  rotation  will  be  measured  by  the  moment  of  inertia  of  the 
section  with  reference  to  this  axis.  The  maximum  tensile  and 
compressive  stresses  will  occur  at  the  fibre  farthest  removed  from 
X'  X'  or  at  M  and  N,  the  stress  at  M  being  tensile  when  the 
direction  of  P  is  upward  to  the  plane  of  the  paper,  and  com- 
pressive when  its  direction  is  downward.  The  centre  of  gravity, 
C,  may  be  located  readily,  by  finding  the  intersection  of  any  pair 
of  gravity  axes.  If  the  section  has  an  axis  of  symmetry,  as  UV, 
Fig.  187,  it  is  necessary  only  to  find  the  axis  at  right  angles  to  UV. 
This  is  most  readily  done  graphically  as  follows:  Divide  the 
section  into  small  areas,  as  shown  by  dotted  lines  at  xx  in  Fig.  187. 
From  the  centre  of  gravity  of  each  area  draw  parallel  lines  ab, 
be,  cd,  preferably  at  right  angles  to  the  known  axis  UV.  In  Fig. 
187  (a),  lay  off  AB,  BC,  etc.,  proportional  to  the  respective  areas 


MACHINE   FRAMES  AND  ATTACHMENTS 


435 


whose  gravity  axes  are  ab,  be,  etc.  Take  any  pole  O  and  draw 
AO,  BO,  etc.  From  any  point  on  ab,  draw  00  indefinitely,  parallel 
to  AO.  From  the  same  point  draw  ob  parallel  to  OB.  From  the 
intersection  of  ob  and  be  draw  co  parallel  to  CO,  and  from  its 
intersection  with  cd,  draw  od  parallel  to  OD.  The  intersection  of 
ao  and  od  locates  the  gravity  axis  XX  (see  also  Art.  120).  It  is 
evident  that  this  method  may  be  applied  when  both  axes  are  un- 
known. 

The  moment  of  inertia  of  the  section  around  X'  X'  may  be 
most  readily  found  by  transforming  the  area  of  the  figure  into 
an  equivalent  figure  with  RP  as  a  base,  as  follows :  Draw  lines, 


\ 


FIG.  187. 


FIG.  188  (b). 


as  X"  X",  parallel  to  X'  X',  and  plot  the  intercepts  made  by  it 
on  the  given  section,  on  each  side  of  CB  as  ordinates  of  an  equi- 
valent section,  shown  in  Fig.  187  by  the  dotted  line  L.  The 
accuracy  of  the  work  may  be  checked  with  a  planimeter,  as  it  is 
evident  that  the  area  of  the  transformed  section  will  be  equal  to 
that  of  the  original.  Divide  this  equivalent  figure  into  approxi- 
mate rectangles,  by  lines  drawn  parallel  to  X'  X' ,  as  shown  at 
r.  Then  the  moment  of  inertia  of  r  around  the  axis  X'  X'  will 
be  its  moment  of  inertia  round  its  own  gravity  axis  parallel  to 
X'  X',  plus  its  area  into  the  square  of  the  distance  between  these 
axes.  The  sum  of  the  moments  of  inertia  of  all  such  small  areas 
will  be  the  required  moment  of  inertia  of  the  section. 


436  MACHINE  DESIGN 

195.  Distribution  of  Metal  in  Frames.  Machine  frames  are 
usually  made  of  castings,  on  account  of  their  complicated  shapes, 
cast  iron  being  the  material  most  used,  while  steel  castings  are  rap- 
idly coming  into  use  for  severe  work.  In  addition  to  the  stresses 
induced  in  the  frame  by  the  energy  transmitted  by  the  machine, 
it  may  also  be  subjected  to  severe  accidental  stresses  due  to  such 
causes  as  shrinkage,  or  the  settling  of  a  part  of  the  foundation. 
Both  these  classes  of  stresses  are,  in  general,  very  complex  and  gen- 
erally beyond  mathematical  analysis,  and  the  problem  must  fre- 
quently be  left  to  the  judgment  of  the  designer,  especially  if  stiffness 
is  a  large  factor.  Economy  in  the  use  of  metal,  however,  demands 
that  its  distribution  throughout  the  frame  shall  be  in  accord 
with  the  best  analysis  possible,  and,  therefore,  the  general  prin- 
ciples  governing  the  forms  of  sections  must  be  kept  in  mind. 

The  most  trying  stresses  to  which  a  frame  may  be  subjected 
are  torsion,  flexure,  or  a  combination  of  these.  It  has  been  noted 
in  Art.  12  that  the  hollow  section  (Fig.  7)  is  most  effective  for 
resisting  torsion,  and,  if  this  be  the  predominating  stress  sections 
such  as  are  shown  in  Fig.  7,  or  modified  sections  as  shown  in  Fig. 
187,  are  correct.  It  was  also  noted  in  Art.  19  (Fig.  10)  that  in 
the  case  of  cast  iron,  or  other  metal  whose  tensile  strength  is 
much  less  than  its  compressive  strength,  a  great  saving  of  material 
is  effected  by  massing  the  metal  on  the  tension  side  as  shown  in 
Fig.  1 88  (a) ;  thus  making  the  tensile  and  compressive  stresses  more 
in  proportion  to  the  strength  of  the  material.  If  then  the  pre- 
dominating stress  in  a  frame  is  simple  flexure  (in  a  given  plane), 
a  section  like  that  shown  in  Fig.  188  (a)  is  allowable,  but  if,  in 
addition,  torsional  strength  must  be  withstood,  or  if  the  plane  of 
flexure  may  change,  a  section  similar  to  that  shown  in  Fig.  188  (b) 
is  better  design,  since  it  combines  the  merits  of  both  Figs.  187  and 
1 88  (a).  Sometimes  it  is  better  to  make  the  section  so  large  that 
the  flexural  stress  can  be  safely  withstood  by  a  wall  of  uniform 
thickness,  as  in  Fig.  187,  as  the  construction  of  the  pattern  is 
simpler  and  the  shrinkage  stresses  less  serious  than  in  such  sec- 
tions as  shown  in  Fig.  188.  .  The  metal  in  the  walls  will  be  much 
sounder,  also,  as  the  thick  sections  of  Fig.  188  are  very  likely  to 
have  a  porous  interior,  due  to  shrinkage.  Cast-iron  parts  more 


MACHINE  FRAMES  AND  ATTACHMENTS  437 

than  four  or  five  inches  thick  are  almost  sure  to  be  defective  in 
this  manner.  The  walls  of  such  sections  as  shown  in  Fig.  188 
should  taper  uniformly  from  the  thick  part  to  the  thin  parts,  and 
all  corners  should  be  well  rounded,  and  filleted,  to  minimize  as 
far  as  possible  the  concentration  of  shrinkage  stresses.  Thin 
wide  flanges  or  webs  should  not  be  cast  integral  with  thick  heavy 
parts,  as  unequal  shrinkage  and  porosity  are  sure  to  result.  This 
is  especially  true  of  thin  ribs  cast  on  the  tension  side  of  large 
sections,  as  the  edge  of  the  rib  is  liable  to  crack  through  shrinkage, 
thus  starting  rupture  across  the  entire  section.  Small  brackets 
or  other  attachments  of  thin  sections  should  never  be  cast  on  a 
large  frame,  as  they  seldom  cast  well.  A  section  of  moderate 
thickness  is  often  stronger  than  a  thicker  one,  since  the  greatest 
strength  of  cast  iron  is  in  the  outer  skin.  It  should  also  be  re- 
membered that  even  when  a  frame  is  both  strong  and  stiff  enough 
to  do  the  required  work  at  low  speeds,  it  may  not  have  mass 
enough  to  absorb  the  vibrations  set  up  when  running  more 
rapidly.  This  may  call  for  more  metal  in  the  frame  than  is 
dictated  by  other  requirements.  Openings  for  supporting  or 
removing  cores  should  be  placed  near  the  gravity  axis  so  as 
to  reduce  the  strength  as  little  as  possible  (see  Fig.  191). 

196.  Attachments  and  Supports.  The  general  appearance  of 
a  machine  is  affected  more  by  the  outline  of  the  main  frame  than 
by  that  of  any  other  member.  This  outline  should,  therefore,  be 
clearly  shown,  and  not  obliterated  at  places  by  the  various  attach- 
ments which  restrain  the  moving  parts  or  support  the  frame. 
In  Fig.  183  is  shown  the  outline  of  a  frame  in  which  the  various 
sections  have  been  proportioned  in  accordance  with  the  loads 
brought  upon  them,  and  the  various  bosses  N  and  the  support  S 
appear  as  attachments  to  the  main  member.  Fig.  184  illustrates 
the  same  machine  with  the  attachments  merged  into  the  main 
member,  thereby  destroying  the  character  of  the  design,  and  also 
making  it  more  difficult  to  judge  of  the  relative  strength  of  various 
sections  of  the  frame. 

The  form  of  an  attachment  will,  of  course,  be  governed  by 
the  service  it  is  required  to  render  and  the  manner  in  which  it 
is  loaded  and  supported.  If  the  outline  of  the  attachment  is 


438 


MACHINE  DESIGN 


based  on  theoretical  considerations,  care  should  be  exercised 
that  all  the  modifying  influences  are  duly  considered.  Thus 
if  parabolic  outlines  are  given  to  an  attachment,  such  as  the 
housings  H  for  supporting  the  tool  in  Fig.  192,  the  upper  end  of 
the  housing  must  be  modified  from  the  theoretical  parabolic 
outline  indicated  by  the  bending  effect  of  the  force  P,  so  as  to 
provide  for  the  shearing  effect  at  the  upper  end,  which  is  fre- 
quently neglected.  (See  also  Article  15.) 

If  the  frame  rests  directly  on  the  floor  its  outlines  should  be 
carried  down  to  the  floor  in  such  a  manner  as  will  give  an  appear- 


FIG.  189. 


FIG.  190. 


ance  of  stability.  Thus  Fig.  189  shows  such  a  machine  frame 
on  which  the  vertical  outline  of  the  back  of  the  frame  is  undercut. 
Fig.  190  shows  the  same  machine  with  the  outline  carried 
straight  to  the  floor  and  the  improvement  in  appearance,  so  far 
as  stability  is  concerned,  is  obvious.  Fig.  191  shows  the  outline 
of  a  planing  machine  in  which  the  upright,  U,  is  carried  to  the 
floor  at  V,  in  the  form  of  a  leg.  This  construction  is  not  correct, 
as  U  is  an  attachment  to  the  bed,  designed  to  resist  the  force  of 
the  cut  and  transfer  it  to  the  bed,  which  should  itself  be  stiff 
enough  to  withstand  all  such  stress  thus  brought  upon  it. 
Any  settling  of  the  foundation  might  affect  the  alignment  of 


MACHINE   FRAMES  AND  ATTACHMENTS 


439 


U  and  hence  the  arrangement  shown  in  Fig.  192  is  more  nearly 
correct. 

In  large  machines  the  frame  usually  rests  directly  on  the 
foundation,  and  should  have  sufficient  stiffness  to  resist  distortion 
due  to  the  settling  of  the  foundation,  since  the  latter  is  very 
difficult  to  avoid.  In  smaller  machines  the  frame  is  carried  on 
supports,  which  may  be  of  two  general  types,  (a)  cabinet  or  box 
pillar  supports  (Fig.  192),  and  (6)  legs  as  shown  in  Fig.  193. 
The  choice  of  support  will,  of  course,  depend  on  the  type  and 
size  of  the  machines.  In  any  case  the  number  of  points  of 
support  should  be  as  few  as  possible.  If  the  machine  can  be 
supported  on  three  points  it  is  evident  that  the  frame  cannot  be 
affected  by  settling  of  the  foundation.  It  is  difficult,  in  general, 


FIG.  192. 


to  obtain  three-point  support,  but  it  is  seldom  necessary  to  place 
supports  as  close  together  as  in  Fig.  191  (which  is  taken  from  an 
actual  design),  where  the  frame  is  carried  on  eight  points.  Fig. 
192  shows  the  same  frame  properly  carried  on  box  supports, 
the  supports  themselves  being  so  stiff  as  materially  to  assist  the 
frame  and  practically  reducing  the  support  to  so-called  two-point 
support.  Small  machines  can  often  be  supported  on  a  single 
box-pillar,  the  overhanging  parts  of  the  frame  having,  a  parabolic 
outline  as  suggested  in  Fig.  189.  If  the  box  pillar  is  of 
considerable  height  the  sides  should  taper  slightly  toward 
the  top;  for  if  made  parallel  the  pillar  will  appear  wider  at 
the  top  than  at  the  bottom.  It  is  preferable  to  use  one  form 
of  support  throughout,  i.e.,  all  box  pillars  or  all  legs,  and  not 
one  or  more  of  each. 


440 


MACHINE  DESIGN 


When  the  frame  must  be  supported  on  legs,  as  in  Fig.  195, 
these  should  not  curve  outward  as  in  Fig.  193,  unless  it  is  abso- 
lutely essential  in  order  to  obtain  stability.  Spreading  the  legs 
as  in  Fig.  193  lengthens  the  distance  between  the  reactions,  R,  R, 
and,  therefore,  increases  the  bending  effect  on  the  bed  and  legs 
as  a  whole.  The  leg  shown  in  profile  in  Fig.  194  is  better  and 
much  easier  to  make.  The  legs  should  be  so  placed  that  the 
outline  L  forms  a  continuation  of  the  principal  vertical  outline  L' 
of  the  frame,  as  shown  in  Fig.  194.  The  same  remarks  apply  to 
the  end  view  of  the  legs  as  shown  in  Figs.  195  and  196.  The 
complex  curves  and  ornate  features  of  Fig.  195  are  not  only  use- 


FIG.  193. 


FIG.  194.        FIG.  195. 


FIG.  196. 


less  but  expensive.  It  is  not  always  possible  or  desirable  to  make 
machine  frames  and  supports  with  simple  straight-line  outlines; 
but  where  curves  are  necessary  they  should  be  as  simple  as 
possible ;  and  in  general  the  best  results  can  be  obtained  by  using 
arcs  of  circles  or  parabolas.  Ornamentation  of  a  fanciful  nature 
is  not  permissible  anywhere,  as  it  really  detracts  from  the  appear- 
ance of  the  machine,  and  adds  to  the  cost  of  production.  Har- 
mony of  design  can  be  attained  by  making  the  various  members 
of  correct  proportions  to  withstand  the  loads  brought  upon 
them,  and  by  using  the  simplest  and  most  direct  design  with 
smooth  transition  curves  between  straight  lines  which  intersect. 
It  is  a  proverb  in  design  that  "  what  is  right  looks  right." 


INDEX 


ABSOLUTE  efficiency,  no 
Accumulator,  hydraulic,  29 
Air  compressor,  26 
Air  reservoir,  29 
Anti-friction  metals,  233 
Apparent  factor  of  safety,  89 
Axles,  285 

BABBITT  metal,  233 
Ball  bearings,  277 

allowable  load  on,  282 
Bands,  thin,  205 

Barnard,  Prof.  W.  N.   (riveted  fasten- 
ings), 149 
Beams,  general  theory  of,  40 

of  uniform  strength,  42 
Bearing  pressures  on  journals,  table  of, 

251 

on  sliding  surfaces,  238 
Bearing,  step,  264 
Bearings,  allowable  pressure  on,  251 

ball,  271,  277 

collar,  264 

construction  of,  243 

forms  of,  239 

metals  for,  233 

perfectly  lubricated,  253 

radiation  of  heat  from,  247 

roller,  271,  273 

table  of  proportions  of,  252 

thrust,  263 
Belt,  example  of  design  of,  314 

transmission,  theory  of,  308 
Belting,  efficiency  of,  318 

weight  of,  314 
Belts,  coefficient  of  friction  of,  313 

construction  of,  308 

creep  of,  310 

practical  consideration  of,  319 

practical  rules  for,  318 


Belts,  slip  of,  310 

velocity  of,  317 

Bending  moment,  equivalent  or  ideal,  49 
Bevel  gears,  383 
Block  brakes,  355 
Boiler  plate,  strength  of,  154 

rivets,  strength  of,  154 
Bolts,  allowable  stress  in,  176 

efficiency  of,  172 

experiments  on  the  strength  of,  170 

for  reinforcing  castings,  207 

initial  tension  in,  169 

location  of,  181 

Professor  Sweet's  experiments  on, 
181 

resilience  of,  178 

resultant  stress  in,  172 

straining  action  in,  168,  169,  172 

stud,  163,  164 

tap,  163,  164 

through,  163,  164 
Brakes,  block,  355 

coefficient  of  friction  for,  363 

differential,  359 

friction,  355 

strap,  357 

Briggs'  system  of  pipe  threads,  168 
Butt  joints  in  plates,  138,  146 

CAP  screws,  163 

Carman,   Prof.  A.  P.,  experiments  on 

tubes,  218 

Carrying  strength,  89 
Chain  drums  and  sheaves,  340 

Rehold,  Morse,  348 

roller,  block,  stud,  345 
Chains,  338 

conveyor,  344 

for  power  transmission,  344 

proof  test  of,  340 


441 


442 


INDEX 


Chains,  silent,  345 

strength  of,  340 

weldless,  340 
Clavarino's  formula,  223 
Clutches,  allowable  pressure  on,  363 

band,  362 

coefficient  of  friction  for,  363 

conical,  359 

disc,  361 

friction,  359 

magnetic,  363 

radially  expanding,  360 

shaft,  301,  305 
Coefficient  of  elasticity,  34 

of  friction  for  screws,  184 
Coefficients  of  friction  for  brakes  and 
clutches,  363 

of  friction  for  friction  wheels,  353 

of  friction  of  pivots,  269 
Collar  bearings,  264,  267 
Columns,  eccentric  loading  of,  73 

or  long  struts,  61 
Compression  and  torsion,  combined,  57 

in  machine  elements,  36 
Conservation  of  energy,  3-6 
Constraining  surfaces,  materials  of,  232 
Continuous  system  of  rope-driving,  329 
Cotters,  stresses  in,  198 
Coupling,  flange  shaft,  303 

Hook's,  304 

Oldham,  304 
Couplings,  flexible  shaft,  306 

shaft,  301 

Crank-effort  diagram,  20 
Cycloidal  gear  teeth,  365 
Cylinders,  thick,  223 

thin,  211,  215 

Deflection  of  ropes,  332 

table  of,  333 

Deformation,  work  of,  77 
Differential  brake,  359 
Discs,  rotating,  425 

Efficiency,  absolute,  no 
definition  of,  6 
general  theory  of,  109 
mechanical,  no 


Efficiency  of  belting,  318 

of  bolts,  172 

of  riveted  fastenings,  141 

of  screws,  157 

of  square-threaded  screws,  157 

of  triangular-threaded  screws,  162 
Efficiencies  of  machine  elements,  113 
Elastic  limit,  34 

resilience,  77 

Elasticity,  coefficient  of,  34 
Energy  cycle,  6 

in  air  compressor,  26 

in  steam  engine,  16 
Energy  problems,  6 

redistribution  of,  29 
Euler's  formula  for  columns,  62 

Factor  of  safety,  35,  88 

on  boiler  work,  155 
Factors  of  safety,  table  of,  91 
Fairbairn,    Sir    Wm.,    experiments    on 

flues,  217 

Fatigue  of  materials,  82 
Feather  keys,  196 

Feathers,  table  of  dimensions  of,  197 
Flanges,  pipe,  226 

Flather,  Prof.,  on  rope  drives,  323,  324 
Flexure  and  direct  stress,  58 
torsion  combined,  43 

in  machine  elements,  40 
Flues,  2ii 

Flywheel  rim  joints,  422 
Flywheels,  406 

coefficients  of  fluctuation,  412 

construction  of,  419 

experiments  on  the  strength  of,  424 

general  theory  of,  406 

stresses  in,  413 
Force  fits,  200 

practical  considerations  in,  204 
stresses  due  to,  201 
Forces  acting  on  machines,  6,  9,  31 
Friction,  applications  of,  350 

clutches,  359 

coefficient  of,  97,  99,  104,  105 

general  theory  of,  96 

laws  of,  98 

of  circular  surfaces,  97 


INDEX 


443 


Friction  of  dry  surfaces,  98 
of  flat  surfaces,  97 
of  lubricated  surfaces,  99 
of  triangular  threads,  162 
of  screws,  157 
of  rolling,  99 
static,  100 

summary  of  general  laws  of,  109 
wheels,  allowable  pressures  on,  352 

coefficients  of  friction  for,  353 

forms  of,  350 

materials  for,  352 

power  transmitted  by,  353 

wedge-faced,  354 
work  of,  97 
Furnace  flues,  corrugated,  222 

GEAR  teeth,  allowable  stresses  in,  386 

cut,  369 

cycloidal,  365 

Fellows  system  of  stub,  390 

Hunt  system,  390 

involute,  365 

machine  moulded,  369 

methods  of  making,  369 

proportions  of,  368,  370 

shrouding  of,  389 

strength  of,  376 

stub,  390 

wear  on,  388 

width  of  face  of,  388 
wheels,  forces  acting  on,  373 

mortise,  371 

rawhide,  372 

strength  of  rims  and  arms,  391 
Gearing,  efficiency  of  spur,  392 
general  principles  of,  364 
helical  or  twisted,  392 
herring-bone,  393 
interchangeable  systems  of,  366 
screw,  395 
skew-bevel,  395 
spiral,  395 

standard  forms  of,  366 
strength  of  twisted,  393 
worm,  395 

Gears,  allowable  speed  of,  387 
bevel,  383 


Gears,  rawhide,  allowable  load  on,  389 
Gordon's  formula  for  columns,  67 

HELICAL  gearing,  392 
Hindley  worm,  398 
Hobs  and  hobbing,  397 
Hoisting  mechanism,  9,  29 
Hook's  coupling,  305 
Hooks,  hoisting,  341 

strength  of,  341 

table  of  proportions  of,  343 
Hoops,  205 

Hunt,  C.  W.,  on  rope  driving,  325 
system  of  gear  teeth,  390 

IMPACT,  shock,  78 
Imperfect  lubrication,  102 
Inertia  effects  in  general,  29 
redistribution  of,  29 
Inertia  forces  in  steam  engines,  19 
Involute  gear  teeth,  365 

JOHNSON'S,  J.  B.,  formula  for  columns, 

66 
T.  H.,  formula  for  columns,  65 

Journals,  bearing  pressure  on,  251 
design  of,  245,  257 
examples  of  design  of,  257 
imperfectly  lubricated,  249 
perfectly  lubricated,  253 

KEYS,  draw,  192 

flat,  190 

forms  of,  190 

saddle,  190 

stresses  in,  192 

sunk,  190 

table  of  dimensions  of  sunk,  196 

Woodruff,  191 
Kinematics,  6 

LAME'S  formula,  223 

Lap  joints  in  plates,  138,  145 

Lasche,  experiments  of,  247,  255 

Launhart's  formula,  85 

Lewis',  Wilfred,  formula  for  gear  teeth, 

376 
Live  load,  effect  of,  82 


444 


INDEX 


Load,  steady,  dead,  suddenly  applied,  31 
Lubrication,  imperfect,  102 

methods  of,  100,  261 

of  journals,  methods  of,  261 

of  sliding  surfaces,  238 

perfect,.  1 06 

MACHINE  attachments,  437 
design,  definition  of,  i 
frames,  428 

distribution  of  metal  in,  436 
stresses  in,  428 
stresses  in  closed,  432 
stresses  in  open,  429 
screws,  163,  164 
supports,  437 

McBride,   James,   experiments  on  effi- 
ciency of  bolts,  172 
Mechanical  advantage,  28 

efficiency,  no 
Mechanism,  definition  of,  2 
Micro  flaws,  theory  of,  83 
Moore,  Prof.  H.  F.,  experiments  of,  107 

on  riveted  fastenings,  149 
Morse  chain,  348 
Multiple  system  of  rope-driving,  329 

OIL  film,  1 06 

in  perfect  lubrication,  106 
grooves,  262 
Oldham  coupling,  304 

PERFECT  lubrication,  106 

Pipe  couplings  and  flanges,  226 

threads,  168 
Pipes,  211,  217 

Piping,  practical  considerations  of,  224 
Pivots,  coefficient  of  friction  of,  269 
Planing  machine,  30 
Plates,  thin,  228 
Power,  definition  of,  7 
Pulleys,  406 
Punching  machine,  10,  60 

RANKINE'S  equation  for  columns,  67 
Relative  strength  of  riveted  fastenings, 

141 
Renold  chain,  348 


Resilience,  76 
elastic,  77 
of  bolts,  178 

Ritter's  formula  for  columns,  68 
Riveted  fastenings,  butt  joints,  138,  146 
chain  riveting,  138 
efficiency  of,  141 
factor  of  safety  in,  155 
failure  of,  141 
forms  of  joints,  137 
general  considerations,  136 
general  equations  for,  147 
lap  joints,  138,  144 
making  of,  151 
marginal  strength  of,  143 
practical  consideration  of,  149 
practical  rules  for,  155 
relative  strength  of,  141 
staggered  riveting,  138 
strength  of  materials  foi,  154 
stresses  in,  139 
theoretical  strength  of,  144 
Riveting,  machine,  153 
Rivets,  diagonal  pitch  of,  138,  143 
pitch  of,  138 

transverse  pitch  of,  138,  143 
Roller  bearings,  271,  273 

allowable  load  on,  276 
Rope-driving,  sheaves  for  fibrous,  331 

systems  of  fibrous,  329 
Rope  transmission,  theory  of,  309,  323 

(by  wire),  338 
Ropes,  cotton,  322 
Ropes,  deflection  of  fibrous,  332 
fibrous  hoisting,  337 
hemp,  leather,  etc.,  322 
Manila,  322 

materials  for  fibrous,  322 
materials  for  wire,  334 
strength  of  fibrous,  327 
strength  of  fibrous  hoisting,  338 
strength  of  wire  hoisting,  339 
velocity  of  fibrous,  327 
wire  hoisting  ropes,  338 
Rotating  discs,  425 

SCREW  fastenings,  163 
gearing,  395 


INDEX 


445 


Screw  and  screw  fastenings,  156 
Screws,  bearing  pressure  on,  186 

cap,  163,  164 

coefficient  of  friction  of,  184 

design  of,  for  power  transmission, 
187 

efficiency  of,  157 

for  power  transmission,  183 

for  power  transmission,   efficiency 
of,  184 

forms  of,  156 

friction  of,  157 

machine,  163,  164 

mechanical  advantage  of,  183 

multiple-threaded,  184 

stresses  in  transmission,  186 

U.  S.  or  Sellers  standard,  166 

Whit  worth  standard,  166 
Sellers  shaft  coupling,  303 

standard  screws,  166 
Set  screws,  163,  165 
Shaft  clutches,  301 

coupling,  flange,  303 

couplings,  301 
Shafts,  allowable  deflection  of,  299 

allowable  span  of,  299 

factors  of  safety  for,  289 

hollow,  300 

subjected  to  torsion,  288 

subjected  to  torsion  and  bending, 
290 

torsional  stiffness  of,  298 

whirling  of,  299 
Shaping   machine,    energy   distribution 

in,  10 

Shear  in  machine  elements,  36 
Shock  in  machine  members,  78 
Shrink  fits,  200,  207 

practical  considerations  in,  204 
Shrouding  of  gear  teeth,  389 
Sliding  surfaces,  233 

bearing  pressures  on,  238 
lubrication  of,  238 
Spheres,  213 
Splines,  196 
Springs,  applications  of,  114 

characteristics  of,  114 

flat,  116 


Springs,  flat,  design  of,  119 
forms  of,  116 
helical,  117 

design  of,  128 
springs  in  torsion,  135 
spiral,  118 

laminated  or  plate,  design  of,  123 
materials  of,  115 
spiral,  118 

Spur  gear  teeth,  strength  of,  376 
gearing,  efficiency  of,  392 
gears,  allowable  speed  of,  387 
allowable  stress  in,  386 
machine-moulded,  369 
width  of  face  of,  388 
Stayed  surfaces,  231 
Steam  engine,  energy  distribution  in,  16 
Step  bearing,  264 
Stewart,  Prof.   R.  T.,   experiments  on 

tubes,  218 
Storage  battery,  29 
Strain,  definition  of,  32 
Straining  action,  nature  of,  32 

table  of  formulae  for,  94 
Strap  brakes,  357 
Strength  of  materials,  table  of,  93 
Stress,  compound,  33,  40 
definition  of,  32 
predominating  or  primary,  40 
strain  diagram,  33 
working,  35 

Stribeck,  Prof.,  experiments  of,  255,  272 
Stub  gear  teeth,  390 
Stud  bolts,  163,  164 
Sweet,  Prof.,  method  of  relieving  sliding 
surfaces,  237 

TAP  bolts,  164 

Taylor,  F.  M.,  rules  for  belting,  319 

Temperature,   coefficient  of  expansion, 

75 

stresses  due  to,  75 
Tension  in  machine  elements,  35 
Thrust  bearing  for  worms,  405 
Thrust  bearings,  263 

allowable  pressures  on,  270 
efficiency  of,  268 
Toothed  gearing,  angular  velocity  of,  365 


446 


INDEX 


Toothed  gearing,  classification  of,  364 
interchangeable     systems     of, 

366 
Torsion    and    compression,    combined, 

57 

and  flexure,  combined,  43 

in  machine  elements,  36 
Tower,  Beaucamp,  experiments  of,  254 
Tower's  experiments,  106 
Towne,  H.  R.,  experiments   on   hooks, 

343 
Triangular  threads,  efficiency  of,  162 

friction  of,  162 
Tubes,  211,  217 
Twisted  gears,  392 

ULTIMATE  strength,  definition  of,  34 

Unions,  pipe,  226 

U-  S.  standard  screws,  table  of,  167 


VAN  STONE  pipe  flanges,  227 

WEYRAUCH'S  formula,  86 
Whit  worth  standard  screws,  166 
Wire  rope  transmission,  333 

theory  of,  334 
ropes,  materials  for,  334 

power  transmitted  by,  336 
Wohler's  experiments  of,  84 
Work  of  deformation,  76 
Working  stress,  35 
Worm  and  worm  wheel,  395 
gearing,  design  of,  403 
efficiency  of,  399 
limiting  pressures  on,  401 
limiting  velocities  of,  401 
velocity  ratio  of,  398 
Hindley,  398 
thrust  bearing,  405 


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Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trautwine.) 8vo,  4  00 

7 


Hazen's  Clean  Water  and  How  to  Get  It Large  12mo,  $1  50 

Filtration  of  Public  Water-supplies 8vo,  3  00 

Hazelhurst's  Towers  and  Tanks  for  Water-works 8vo,  2  50 

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*  Thomas  and  Watt's  Improvement  of  Rivers 4to,  6  00 

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Whipple's  Value  of  Pure  Water Large  12mo,  1  00 

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Wood '  s  Turbines 8 vo ,  2  50 


MATERIALS    OF    ENGINEERING. 

Baker's  Roads  and  Pavements 8vo,  5  00 

Treatise  on  Masonry  Construction 8vo,  5  00 

Black's  United  States  Public  Works Oblong  4to,  5  00 

Blanchard's  Bituminous  Roads.      (In  Press.) 

Bleininger's  Manufacture  of  Hydraulic  Cement.      (In  Preparation.) 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  50 

Byrne's  Highway  Construction 8vo,  5  00 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

16mo,  3  00 

Church's  Mechanics  of  Engineering 8vo,  6  00 

Du  Bois's  Mechanics  of  Engineering. 

Vol.    I.  Kinematics,  Statics,  Kinetics Small  4to,  7  50 

Vol.  II.  The  Stresses  in  Framed  Structures,  Strength  of  Materials  and 

Theory  of  Flexures Small  4to,  10  00 

*  Eckel's  Cements,  Limes,  and  Plasters 8vo,  6  00 

Stone  and  Clay  Products  used  in  Engineering.     (In  Preparation.) 

Fowler's  Ordinary  Foundations 8vo,  3  50 

*  Greene's  Structural  Mechanics 8vo,  2  50 

*  Holley's  Lead  and  Zinc  Pigments Large  12mo,  3  00 

Holley  and  Ladd's  Analysis  of  Mixed  Paints,  Color  Pigments  and  Varnishes. 

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Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Maire's  Modern  Pigments  and  their  Vehicles 12mo,  2  00 

Martens's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo,  750 

Maurer's  Technical  Mechanics 8vo,  4  00 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  00 

Merriman's  Mechanics  of  Materials 8vo,  5  00 

*  Strength  of  Materials 12mo,  1  00 

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Pattern's  Practical  Treatise  on  Foundations 8vo,  6  00 

Rice's  Concrete  Block  Manufacture 8vo,  2  00 

8 


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States 8vo.  2  50 

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Smith's  Strength  of  Material 12mo, 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement 12mo,  2  00 

Text-book  on  Roads  and  Pavements 12mo,  2  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  00 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  00 

Part  I.      Non-metallic  Materials  of  Engineering  and  Metallurgy.. .  ,8vo,  2  00 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.    A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  00 

Turneaure  and  Maurer's  Principles  of  Reinforced  Concrete  Construction. 

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Waterbury's  Cement  Laboratory  Manual 12mo,  1  00 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  00 

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Steel 8vo,  4  00 


RAILWAY   ENGINEERING. 

Andrews's  Handbook  for  Street  Railway  Engineers. 3X5  inches,  mor.  1  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  00 

Brooks's  Handbook  of  Street  Railroad  Location 16mo,  mor.  1  50 

Butts's  Civil  Engineer's  Field-book 16mo,  mor.  2  50 

Crandall's  Railway  and  Other  Earthwork  Tables 8vo,  1  50 

Transition  Curve 16mo,  mor.  1  50 

*  Crockett's  Methods  for  Earthwork  Computations 8vo,  *   50 

Dredge's  History  of  the  Pennsylvania  Railroad.   (1879) Papei  5  00 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide. .  16mo,  mor.  2  50 
Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
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Ives  and  Hilts's  Problems  in  Surveying,  Railroad  Surveying  and  Geodesy 

16mo,  mor.  1  50 

Molitor  and  Beard's  Manual  for  Resident  Engineers 16mo,  1  00 

Nagle's  Field  Manual  for  Railroad  Engineers 16mo,  mor.  3  00 

*  Orrock's  Railroad  Structures  and  Estimates 8vo,  3  00 

Philbrick's  Field  Manual  for  Engineers 16mo,  mor.  3  00 

Raymond's  Railroad  Engineering.     3  volumes. 

Vol.      I.  Railroad  Field  Geometry.      (In  Preparation.) 

Vol.    II.  Elements  of  Railroad  Engineering 8vo,  3  50 

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Searles's  Field  Engineering 16mo,  mor.  3  00 

Railroad  Spiral 16mo,  mor.  1  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  1  50 

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12mo,  mor.  2  50 

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Railroad  Construction 16mo,  mor.  5  00 

Wellington's  Economic  Theory  of  the  Location  of  Railways Large  12mo,  5  00 

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9 


DRAWING. 

Ban's  Kinematics  of  Machinery 8vo,  $2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  00 

*  "                                                      Abridged  Ed 8vo,  150 

Coolidge's  Manual  of  Drawing 8vo,  paper,  1  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to,  2  50 

Durley's  Kinematics  of  Machines 8vo,  4  00 

Emch's  Introduction  to  Projective  Geonaetry  and  its  Application 8vo,  2  50 

French  and  Ives'  Stereotomy 8vo,  2  50 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,  2  00 

Jamison's  Advanced  Mechanical  Drawing 8vo,  2  00 

Elements  of  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

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Kimball  and  Barr's  Machine  Design.     (In  Press.) 

MacCord's  Elements  of  Descritpive  Geometry 8vo,  3  00 

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Mechanical  Drawing 4to,  4  00 

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Reed's  Topographical  Drawing  and  Sketching 4to,  5  00 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design.. 8vo,  3  00 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  00 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  00 

Smith's  (R.  S.)  Manual  of  Topographical  Drawing.  (McMillan) 8vo,  2  50 

*  Titsworth's  Elements  of  Mechanical  Drawing Oblong  8vo,  1  25 

,  Warren's  Drafting  Instruments  and  Operations 12mo,  1  25 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  50 

Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  . .  .  12mo,  1  00 

General  Problems  of  Shades  and  Shadows 8vo,  3  00 

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Shadow 12mo,  1  00 

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Weisbach's     Kinematics    and    Power    of    Transmission.      (Hermann    and 

Klein.) » 8vo,  5  00 

Wilson's  (H.  M.)  Topographic  Surveying ; 8vo,  3  50 

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Woolf 's  Elementary  Course  in  Descriptive  Geometry Large  8vo,  3  00 


ELECTRICITY  AND   PHYSICS. 

*  Abegg's  Theory  of  Electrolytic  Dissociation,     (von  Ende.) 12mo,  1  25 

Andrews's  Hand-book  for  Street  Railway  Engineering 3X5  inches,  mor.  1  25 

Anthony  and  Brackett's  Text-book  of  Physics.      (Magie.) Large  12 mo,  3  00 

Anthony  and   Ball's  Lecture-notes  on  the  Theory  of  Electrical  Measure- 
ments  12mo,  1  00 

Benjamin's  History  of  Electricity , 8vo,  3  00 

Voltaic  Cell 8vo,  3  00 

10 


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Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).Svo,  3  00 

*  Collins's  Manual  of  Wireless  Telegraphy  and  Telephony 12mo,  1  50 

*  Mor.  2  00 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo,  3  00 

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Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  . .  .  16mo,  mor.  5  00 
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Duhem's  Thermodynamics  and  Chemistry.      (Burgess.) 8vo,  4  00 

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Getman's  Introduction  to  Physical  Science 12mo, 

Gilbert's  De  Magnete.      (Mottelay) 8vo,  2  50 

*  Hanchett's  Alternating  Currents 12mo,  1  00 

Hering's  Ready  Reference  Tables  (Conversion  Factors) 16mo,  mor.  2  50 

*  Hobart  and  Ellis's  High-speed  Dynamo  Electric  Machinery 8vo,  6  00 

Holman's  Precision  of  Measurements 8vo,  2  00 

Telescopic  Mirror-scale  Method,  Adjustments,  and  Tests..  .  .Large  8vo,  75 

*  Karapetoff 's  Experimental  Electrical  Engineering 8vo.  6  00 

Kinzbrunner's  Testing  of  Continuous-current  Machines 8vo,  2  00 

Landauer's  Spectrum  Analysis.      (Tingle.) 8vo,  3  00 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess. )12mo,  3  00 

Lob's  Electrochemistry  of  Organic  Compounds.      (Lorenz) 8vo,  3  00 

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*  Lyons's  Treatise  on  Electromagnetic  Phenomena.  Vols,  I  .and  II.  8vo,  each,  6  00 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,  4  00 

Morgan's  Outline  of  the  Theory  of  Solution  and  its  Results 12mo,  1  00 

*  Physical  Chemistry  for  Electrical  Engineers 12mo,  1  50 

*  Norris's  Introduction  to  the  Study  of  Electrical  Engineering 8vo,  2  50 

Norris  and  Denmson's  Course  of  Problems  on  the  Electrical  Characteristics  of 

Circuits  and  Machines.      (In  Press.) 

*  Parshall  and  Hobart's  Electric  Machine  Design 4to,   half  mor,  12  50 

Reagan's  Locomotives:  Simple,  Compound,  and  Electric.     New  Edition. 

Large  12mo,  3  50 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.) .  .8vo,  2  00 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Schapper's  Laboratory  Guide  for  Students  in  Physical  Chemistry 12mo,  1  00 

*  Tillman's  Elementary  Lessons  in  Heat. 8vo,  1  50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Large  12mo,  2  00 

Ulke's  Modern  Electrolytic  Copper  Refining * 4 8vo,  3  00 


LAW. 

*  Brennan's  Hand-book  of  Useful  Legal  Information  for  Business  Men. 

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Manual  for  Courts-martial. 16mo,  mor.  1  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep,  6  50 

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Law  of  Operations  Preliminary  to    Construction  in  Engineering  and 

Architecture. 8vo,  5  00 

Sheep,  5  50 


MATHEMATICS. 

Baker's  Elliptic  Functions ,8vo,     I  50 

Briggs's  Elements  of  Plane  Analytic  Geometry.      (B6cher) 12mo,     1  00 

*  Buchanan's  Plane  and  Spherical  Trigonometry 8vo,     1  00 

11 


Byerley's  Harmonic  Functions 8vo,  $1  00 

Chandler's  Elements  of  the  Infinitesimal  Calculus 12mo,  2  00 

*  Coffin's  Vector  Analysis 12mo,  2  50 

Compton's  Manual  of  Logarithmic  Computations 12mo,  1  50 

*  Dickson's  College  Algebra Large  12mo,  1  50 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  12mo,  1  25 

Emch's  Introduction  to  Protective  Geometry  and  its  Application 8vo,  2  50 

Fiske's  Functions  of  a  Complex  Variable 8vo,  1  00 

Halsted's  Elementary  Synthetic  Geometry 8vo,  1  50 

Elements  of  Geometry 8vo,  1  75 

*  Rational  Geometry 12mo,  1  50 

Hyde's  Grassmann's  Space  Analysis 8vo,  1  00 

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100  copies,  5  00 

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10  copies,  2  00 
Johnson's  (W.  W.)  Abridged  Editions  of  Differential  and  Integral  Calculus. 

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Curve  Tracing  in  Cartesian  Co-ordinates 12mo,  1  00 

Differential  Equations f 8vo,  1  00 

Elementary  Treatise  on  Differential  Calculus Large  12mo,  1  50 

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*  Theoretical  Mechanics. 12mo,  3  00 

Theory  of  Errors  and  the  Method  of  Least  Squares 12mo,  1  50 

Treatise  on  Differential  Calculus Large  12mo,  3  00 

Treatise  on  the  Integral  Calculus Large  12mo,  3  00 

Treatise  on  Ordinary  and  Partial  Differential  Equations.  .  .Large  12mo,  3  50 

Karapetoff's  Engineering  Applications  of  Higher  Mathematics. 

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Laplace's  Philosophical  Essay  on  Probabilities.  (Truscott  and  Emory.) .  12mo,  2  00 

*  Ludlow  and  Bass's  Elements  of  Trigonometry  and  Logarithmic  and  Other 

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*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,  1  00 

Macfarlane's  Vector  Analysis  and  Quaternions 8vo,  1  00 

McMahon's  Hyperbolic  Functions 8vo,  1  00 

Manning's  Irrational  Numbers  and  their  Representation  by  Sequences  and 

Series 12mo,  1  25 

Mathematical   Monographs.     Edited  by   Mansfield   Merriman   and   Robert 

S.  Woodward Octavo,  each  1  00 

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No.  2.  Synthetic  Projective  Geometry,  by  George  Bruce  Halsted. 
No.  3.  Determinants,  by  Laenas  Gifford  Weld.  No.  4.  Hyper- 
bolic Functions,  by  James  McMahon.  No.  5.  Harmonic  Func- 
tions, by  William  E.  Byerly.  No.  6.  Grassmann's  Space  Analysis, 
by  Edward  W.  Hyde.  No.  7.  Probability  and  Theory  of  Errors, 
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Maurer's  Technical  Mechanics 8vo,  4  00 

Merriman's  Method  of  Least  Squares 8vo,  2  00 

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Variable 8vo,  2  00 

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Weld's  Determinants 8vo,  1  00 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,  2  00 

Woodward's  Probability  and  Theory  of  Errors 8vo,  1  00 


12 


MECHANICAL   ENGINEERING. 

MATERIALS   OF   ENGINEERING,  STEAM-ENGINES   AND    BOILERS. 

Bacon's  Forge  Practice 12mo,  $1  50 

Baldwin's  Steam  Heating  for  Buildings 12mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

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*  "                                       "       Abridged  Ed 8vo,  1  50 

*  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal 8vo,  3  50 

Carpenter's  Experimental  Engineering 8vo,  6  00 

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Clerk's  Gas  and  Oil  Engine.      (New  edition  in  press.) 

Compton's  First  Lessons  in  Metal  Working 12mo,  1  50 

Compton  and  De  Groodt's  Speed  Lathe 12mo,  1  50 

Coolidge's  Manual  of  Drawing 8vo,  paper,  1  00 

Coolidge  and  Freeman's  Elements  of  Geenral  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

Cromwell's  Treatise  on  Belts  and  Pulleys 12mo,  1  50 

Treatise  on  Toothed  Gearing 12mo,  1  50 

Dingey's  Machinery  Pattern  Making 12mo,  2  00 

Durley's  Kinematics  of  Machines 8vo,  4  00 

Flanders's  Gear-cutting  Machinery Large  12mo,  3  CO 

Flather's  Dynamometers  and  the  Measurement  of  Power 12mo,  3  00 

Rope  Driving 12mo,  2  00 

Gill's  Gas  and  Fuel  Analysis  for  Engineers 12mo,  1  25 

Goss's  Locomotive  Sparks 8vo,  2  00 

Greene's  Pumping  Machinery.      (In  Preparation.) 

Hering's  Ready  Reference  Tables  (Conversion  Factors) 16mo,  mor.  2  50 

*  Hobart  and  Ellis's  High  Speed  Dynamo  Electric  Machinery 8vo,  6  00 

Button's  Gas  Engine 8vo,  5  00 

Jamison's  Advanced  Mechanical  Drawing < 8vo,  2  00 

Elements  of  Mechanical  Drawing 8vo,  2  50 

Jones's  Gas  Engine 8vo,  4  00 

Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  1  50 

Part  I}.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  00 

Kent's  Mechanical  Engineer's  Pocket-Book 16mo,  mor.  5  00 

Kerr's  Power  and  Power  Transmission 8vo,  2  00 

Kimball  and  Barr's  Machine  Design.      (In  Press.) 

Levin's  Gas  Engine.      (In  Press.) 8vo, 

Leonard's  Machine  Shop  Tools  and  Methods 8vo,  4  00 

*  Lorenz's  Modern  Refrigerating  Machinery.   (Pope,  Haven,  and  Dean).  .8vo,  4  00 
MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  00 

Mechanical  Drawing. 4to,  4  00 

Velocity  Diagrams 8vo,  1  50 

MacFarland's  Standard  Reduction  Factors  for  Gases 8vo,  1  50 

Mahan's  Industrial  Drawing.      (Thompson.). 8vo,  3  50 

Mehrtens's  Gas  Engine  Theory  and  Design Large  12mo,  2  50 

Oberg's  Handbook  of  Small  Tools Large  12mo,  3  00 

*  Parshall  and  Hobart's  Electric  Machine  Design.  Small  4to,  half  leather,  12  50 

Peele's  Compressed  Air  Plant  for  Mines 8vo,  3  00 

Poole's  Calorific  Power  of  Fuels 8vo,  3  00 

*  Porter's  Engineering  Reminiscences,  1855  to  1882 8vo,  3  00 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  00 

Richards's  Compressed  Air 12mo,  1  50 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  00 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  00 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  00 

Sorel's  Carbureting  and  Combustion  in  Alcohol  Engines.     (Woodward  and 

Preston.) Large  12mo,  3  00 

Stone's  Practical  Testing  of  Gas  arid  Gas  Meters 8vo,  3  50 

13 


Thurston's  Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

12mo,  $1  00 

Treatise  on  Friction  and  Lost  Work  in  Machinery  and  Mill  Work .  .  .  8vo,  3  00 

*  Tillson's  Complete  Automobile  Instructor 16mo,  1  50 

*  Titsworth's  Elements  of  Mechanical  Drawing Oblong  8vo,  1  25 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

*  Waterbury's  Vest  Pocket  Hand-book  of  Mathematics  for  Engineers. 

21  X  5f  inches,  mor.  1  00 
Weisbach's    Kinematics    and    the    Power    of   Transmission.      (Herrmann — 

Klein.) 8vo,  5  00 

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Wood's  Turbines.  .        8vo,  2  50 


MATERIALS    OF   ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  00 

*  Greene's  Structural  Mechanics 8vo,  2  50 

*  Holley's  Lead  and  Zinc  Pigments Large  12mo  3  00 

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Steels,  Steel-Making  Alloys  and  Graphite Large  12mo,  3  00 

Johnson's  (J.  B.)  Materials  of  Construction 8vo,  6  00 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Maire's  Modern  Pigments  and  their  Vehicles 12mo,  2  00 

Martens's  Handbook  on  Testing  Materials.      (Henning.) 8vo:  7  50 

Maurer's  Techincal  Mechanics 8vo,  4  00 

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*  Strength  of  Materials 12mo,  1  00 

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Sabin's  Industrial  and  Artistic  Technology  of  Paint  and  Varnish 8vo,  3  00 

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Part  II.     Iron  and  Steel 8vo,  3  50 

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Constituents 8vo,  2  50 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  00 

Treatise  on    the    Resistance   of    Materials    and    an    Appendix   on    the 

Preservation  of  Timber 8vo,  2  00 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel. 8vo,  4  00 


STEAM-ENGINES    AND   BOILERS. 

Berry's  Temperature-entropy  Diagram 12mo,  2  00 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.      (Thurston.) 12mo,  1  50 

Chase's  Art  of  Pattern  Making 12mo,  2  50 

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Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  ..  .16mo,  mor.  5  00 

Ford's  Boiler  Making  for  Boiler  Makers 18mo,  1  00 

*  Gebhardt's  Steam  Power  Plant  Engineering 8vo,  6  OO 

Goss's  Locomotive  Performance 8vo,  i  00 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy 12mo.  2  00 

Hutton's  Heat  and  Heat-engines 8vo,  5  00 

Mechanical  Engineering  of  Power  Plants 8vo,  5  00 

Kent's  Steam  boiler  Economy. . 8vo,  4  00 

14 


Kneass's  Practice  and  Theory  of  the  Injector 8vo,  $1  50 

MacCord's  Slide-valves 8vo,  2  00 

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Peabody's  Manual  of  the  Steam-engine  Indicator 12mo,  1  50 

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Valve-gears  for  Steam-engines 8vo.  2  50 

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Thurston's  Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indi- 
cator and  the  Prony  Brake 8vo,  5  00 

Handy  Tables 8vo,  1  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation  8vo,  5  00 

Manual  of  the  Steam-engine 2vols..  8vo.  10  00 

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Wehrenfenning's  Analysis  and  Softening  of  Boiler  Feed-water.     (Patterson). 

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Weisbach's  Heat,  Steam,  and  Steam-engines.      (Du  Bois.) 8vo.  5  00 

Whitham's  Steam-engine  Design 8vo,  5  00 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  .  .8vo,  4  00 


MECHANICS   PURE  AND    APPLIED. 

Church's  Mechanics  of  Engineering 8vo,  6  00 

Notes  and  Examples  in  Mechanics 8vo,  2  00 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools  .12mo,  1  50 
Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.    I.     Kinematics 8vo,  3  50 

Vol.  II.     Statics 8vo,  4  00 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

Vol.  II Small  4to,  10  00 

*  Greene's  Structural  Mechanics 8vo,  2  50 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Large  12mo,  2  00 

*  Johnson's  (W.  W.)  Theoretical  Mechanics 12mo,  3  00 

Lanza's  Applied  Mechanics 8vo,  7  50 

*  Martin's  Text  Book  on  Mechanics.  Voi.  I,  Statics 12mo,  1  25 

*  Vol.  II,  Kinematics  and  Kinetics.  12mo.  1  50 

Maurer's  Technical  Mechanics 8vo.  4  00 

*  Merriman's  Elements  of  Mechanics 12mo,  1  00 

Mechanics  of  Materials > 8vo,  5  00 

••*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  00 

Robinson's  Principles  of  Mechanism , .8vo,  .  3  00 

Sanborn's  Mechanics  Problems Large  12mo,  1  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  00 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  00 

Principles  of  Elementary  Mechanics 12mo,  I  25 


15 


MEDICAL. 

*  Abderhalden's  Physiological  Chemistry  in   Thirty  Lectures.     (Hall  and 

Defren.) 8vo,  $5  00 

von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) 12mo,  1  00 

Bolduan's  Immune  Sera 12mo,  1  50 

Bordet's  Studies  in  Immunity.      (Gay).      (In  Press.) 8vo, 

Davenport's  Statistical  Methods  with  Special  Reference  to  Biological  Varia- 
tions  16mo,  mor.  1  50 

Ehrlich's  Collected  Studies  on  Immunity.     (Bolduan.) 8vo,  6  00 

*  Fischer's  Physiology  of  Alimentation Large  12mo,  2  00 

de  Fursac's  Manual  of  Psychiatry.      (Rosanoff  and  Collins.).. .  .Large  12mo,  2  50 

Hammarsten's  Text-book  on  Physiological  Chemistry.      (Mandel.) 8vo,  4  00 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry .  .8vo,  1  25 

Lassar-Cohn's  Practical  Urinary  Analysis.      (Lorenz.) 12mo,  1  00 

Mandel's  Hand-book  for  the  Bio-Chemical  Laboratory 12mo,  1  50 

*  Pauli's  Physical  Chemistry  in  the  Service  of  Medicine.      (Fischer.)  ..12mo,  1  25 

*  Pozzi-Escot's  Toxins  and  Venoms  and  their  Antibodies.     (Cohn.).  .  12mo,  1  00 

Rostoski's  Serum  Diagnosis.     (Bolduan.) 12mo,  1  00 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  00 

Whys  in  Pharmacy 12mo,  1  00 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Orndorff.)  8vo,  250 

*  Satterlee's  Outlines  of  Human  Embryology 12mo,  1  25 

Smith's  Lecture  Notes  on  Chemistry  for  Dental  Students 8vo,  2  50 

*  Whipple's  Tyhpoid  Fever Large  12mo,  3  00 

Woodhull's  Notes  on  Military  Hygiene 16mo,  1  50 

*  Personal  Hygiene 12mo,  1  00 

Worcester  and  Atkinson's  Small  Hospitals  Establishment  and  Maintenance, 
and  Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small 

Hospital 12mo,  1  25 


METALLURGY. 

Betts's  Lead  Refining  by  Electrolysis 8vo,  4  00 

Bolland's  Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  used 

in  the  Practice  of  Moulding 12mo,  3  00 

Iron  Founder 12mo,  2  50 

Supplement 12mo,  2  50 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo,  1  00 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor.  3  00 

*  Iles's  Lead-smelting 12mo,  2  50 

Johnson's    Rapid    Methods   for    the  Chemical   Analysis   of   Special   Steels, 

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Keep's  Cast  Iron 8vo,  2  50 

Le  Chatelier's  High-temperature  Measurements.     (Boudouard — Burgess.) 

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Metcalf 's  Steel.      A  Manual  for  Steel-users 12mo.  2  00 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.).  .  12mo,  2  50 

Ruer's  Elements  of  Metallography.      (Mathewson) 8vo. 

Smith's  Materials  of  Machines 12mo,  1  00 

Tate  and  Stone's  Foundry  Practice 12mo,  2  00 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  00 

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page  9. 

Part  II.     Iron  and  Steel 8vo,  3  60 

Part  III.  A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  00 

West's  American  Foundry  Practice 12mo,  2  50 

Moulders'  Text  Book 12mo,  2  50 

16 


MINERALOGY. 

Baskerville's  Chemical  Elements.     (In  Preparation.). 

Boyd's  Map  of  Southwest  Virginia Pocket-book  form.  $2  00 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo,  1  50 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo,  4  00 

Butler's  Pocket  Hand-book  of  Minerals 16mo,  mor.  3  00 

Chester's  Catalogue  of  Minerals 8vo,  paper,  1  00 

Cloth,  1  25 

*  Crane's  Gold  and  Silver 8vo,  5  00 

Dana's  First  Appendix  to  Dana's  New  "System  of  Mineralogy".  .Large  8vo,  1  00 
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Large  8vo, 

Manual  of  Mineralogy  and  Petrography 12mo,  2  00 

Minerals  and  How  to  Study  Them 12mo,  1  50 

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Text-book  of  Mineralogy 8vo,  4  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo,  1  00 

Eakle's  Mineral  Tables 8vo,  1  25 

Eckel's  Stone  and  Clay  Products  Used  in  Engineering.      (In  Preparation). 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor.  3  00 

Groth's  Introduction  to  Chemical  Crystallography  (Marshall) 12mo,  1  25 

*  Hayes's  Handbook  for  Field  Geologists 16mo,  mor.  1  50 

Iddings's  Igneous  Rocks 8vo,  5  00 

Rock  Minerals 8vo,  5  00 

Johannsen's  Determination  of  Rock-forming  Minerals  in  Thin  Sections.  8vo, 

With  Thumb  Index  5  00 

*  Martin's  Laboratory     Guide    to    Qualitative    Analysis    with    the    Blow- 

pipe  12mo,  60 

Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses 8vo,  4  00 

Stones  for  Building  and  Decoration 8vo,  5  00 

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Domestic  Production 8vo,  1  00 

*  Pirsson's  Rocks  and  Rock  Minerals 12mo,  2  50 

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*  Ries's  Clays :  Their  Occurrence,  Properties  and  Uses 8vo,  5  00 

*  Ries  and  Leigh  ton's  History  of  the  Clay- working  Industry  of  the  United 

States 8vo,  2  50 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks 8vo,  2  00 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks.  ,  ,,,,,, 8vo,  2  00 


MINING. 

*  Beard's  Mine  Gases  and  Explosions Large  12mo,  3  00 

Boyd's  Map  of  Southwest  Virginia Pocket-book  form,  2  00 

*  Crane's  Gold  and  Silver 8vo,  5  00 

*  Index  of  Mining  Engineering  Literature 8vo,  4  00 

*  8vo,  mor.  5  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo,  1  00 

Eissler's  Modern  High  Explosives 8vo,  4  00 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor.  3  00 

Ihlseng's  Manual  of  Mining 8vo,  5  00 

*  Iles's  Lead  Smelting 12mo,  2  50 

Peele's  Compressed  Air  Plant  for  Mines 8vo,  3  00 

Riemer's  Shaft  Sinking  Under  Difficult  Conditions.      (Corning  and  Peele).8vo,  3  00 

*  Weaver's  Military  Explosives 8vo,  3  00 

Wilson's  Hydraulic  and  Placer  Mining.     2d  edition,  rewritten 12mo,  2  50 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation 12mo,  1  25 

17 


SANITARY   SCIENCE. 

Association  of  State  and  National  Food  and  Dairy  Departments,  Hartford 

Meeting,  1906 8vo,  $3  00 

Jamestown  Meeting,  1907 8vo,  3  00 

*  Bashore's  Outlines  of  Practical  Sanitation 12mo,  1  25 

Sanitation  of  a  Country  House 12mo,  1  00 

Sanitation  of  Recreation  Camps  and  Parks 12mo,  1  00 

Folwell's  Sewerage.      (Designing,  Construction,  and  Maintenance.) 8vo,  3  00 

Water-supply  Engineering 8vo,  4  00 

Fowler's  Sewage  Works  Analyses .  .• 12mo,  2  00 

Fuertes's  Water-filtration  Works 12mo,  2  50 

Water  and  Public  Health 12mo,  1  50 

Gerhard's  Guide  to  Sanitary  Inspections 12mo,  1  50 

*  Modern  Baths  and  Bath  Houses 8vo,  3  00 

Sanitation  of  Public  Buildings 12mo,  1  50 

Hazen's  Clean  Water  and  How  to  Get  It., Large  12mo,  1  50 

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Kinnicut,  Winslow  and  Pratt's  Purification  of  Sewage.     (In  Preparation.) 
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Control 8vo,  7  50 

Mason's  Examination  of  Water.     (Chemical  and  Bacteriological) 12mo,  1  25 

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Ogden's  Sewer  Design 12mo,  2  00 

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Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
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*  Price's  Handbook  on  Sanitation 12mo,  1  50 

Richards's  Cost  of  Cleanness 12mo,  1  00 

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Rideal's  Disinfection  and  the  Preservation  of  Food.  .  .  . , 8vo,  4  00 

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MISCELLANEOUS. 

Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo.  1  50 

Fen-el's  Popular  Treatise  on  the  Winds 8vo,  4  00 

Fitzgerald's  Boston  Machinist 18mo,  1  00 

Gannett's  Statistical  Abstract  of  the  World 24mo,  75 

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